Calculus Examples

$\sin (3 x) d x + 2 y \cos^{3} (3 x) d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = \sin (3 x)$ .

Tap for more steps...

Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [\sin (3 x)]$

Step 1.2

Since $\sin (3 x)$ is constant with respect to $y$ , the derivative of $\sin (3 x)$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = 0$

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = 2 y \cos^{3} (3 x)$ .

Tap for more steps...

Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [2 y \cos^{3} (3 x)]$

Step 2.2

Since $2 y$ is constant with respect to $x$ , the derivative of $2 y \cos^{3} (3 x)$ with respect to $x$ is $2 y \frac{d}{d x} [\cos^{3} (3 x)]$ .

$\frac{\partial N}{\partial x} = 2 y \frac{d}{d x} [\cos^{3} (3 x)]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \cos (3 x)$ .

Tap for more steps...

Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $\cos (3 x)$ .

$\frac{\partial N}{\partial x} = 2 y (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [\cos (3 x)])$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$\frac{\partial N}{\partial x} = 2 y (3 {u_{1}}^{2} \frac{d}{d x} [\cos (3 x)])$

Step 2.3.3

Replace all occurrences of $u_{1}$ with $\cos (3 x)$ .

$\frac{\partial N}{\partial x} = 2 y (3 \cos^{2} (3 x) \frac{d}{d x} [\cos (3 x)])$

Step 2.4

Multiply $3$ by $2$ .

$\frac{\partial N}{\partial x} = 6 y (\cos^{2} (3 x) \frac{d}{d x} [\cos (3 x)])$

Step 2.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 3 x$ .

Tap for more steps...

Step 2.5.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$\frac{\partial N}{\partial x} = 6 y \cos^{2} (3 x) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [3 x])$

Step 2.5.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$\frac{\partial N}{\partial x} = 6 y \cos^{2} (3 x) (- \sin (u_{2}) \frac{d}{d x} [3 x])$

Step 2.5.3

Replace all occurrences of $u_{2}$ with $3 x$ .

$\frac{\partial N}{\partial x} = 6 y \cos^{2} (3 x) (- \sin (3 x) \frac{d}{d x} [3 x])$

Step 2.6

Differentiate.

Tap for more steps...

Step 2.6.1

Multiply $- 1$ by $6$ .

$\frac{\partial N}{\partial x} = - 6 y \cos^{2} (3 x) (\sin (3 x) \frac{d}{d x} [3 x])$

Step 2.6.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{\partial N}{\partial x} = - 6 y \cos^{2} (3 x) \sin (3 x) (3 \frac{d}{d x} [x])$

Step 2.6.3

Multiply $3$ by $- 6$ .

$\frac{\partial N}{\partial x} = - 18 y \cos^{2} (3 x) \sin (3 x) \frac{d}{d x} [x]$

Step 2.6.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{\partial N}{\partial x} = - 18 y \cos^{2} (3 x) \sin (3 x) \cdot 1$

Step 2.6.5

Multiply $- 18$ by $1$ .

$\frac{\partial N}{\partial x} = - 18 y \cos^{2} (3 x) \sin (3 x)$

Step 3

Check that $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ .

Tap for more steps...

Step 3.1

Substitute $0$ for $\frac{\partial M}{\partial y}$ and $- 18 y \cos^{2} (3 x) \sin (3 x)$ for $\frac{\partial N}{\partial x}$ .

$0 = - 18 y \cos^{2} (3 x) \sin (3 x)$

Step 3.2

Since the left side does not equal the right side, the equation is not an identity.

$0 = - 18 y \cos^{2} (3 x) \sin (3 x)$ is not an identity.

Step 4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} d x}$ .

Tap for more steps...

Step 4.1

Substitute $0$ for $\frac{\partial M}{\partial y}$ .

$\frac{0 - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $- 18 y \cos^{2} (3 x) \sin (3 x)$ for $\frac{\partial N}{\partial x}$ .

$\frac{0 - (- 18 y \cos^{2} (3 x) \sin (3 x))}{N}$

Step 4.3

Substitute $2 y \cos^{3} (3 x)$ for $N$ .

Tap for more steps...

Step 4.3.1

Substitute $2 y \cos^{3} (3 x)$ for $N$ .

$\frac{0 - (- 18 y \cos^{2} (3 x) \sin (3 x))}{2 y \cos^{3} (3 x)}$

Step 4.3.2

Cancel the common factor of $0 - (- 18 y \cos^{2} (3 x) \sin (3 x))$ and $2$ .

Tap for more steps...

Step 4.3.2.1

Reorder terms.

$\frac{- 18 \cdot - 1 y \cos^{2} (3 x) \sin (3 x) + 0}{2 y \cos^{3} (3 x)}$

Step 4.3.2.2

Factor $2$ out of $- 18 \cdot - 1 y \cos^{2} (3 x) \sin (3 x)$ .

$\frac{2 (- 9 \cdot - 1 y \cos^{2} (3 x) \sin (3 x)) + 0}{2 y \cos^{3} (3 x)}$

Step 4.3.2.3

Factor $2$ out of $0$ .

$\frac{2 (- 9 \cdot - 1 y \cos^{2} (3 x) \sin (3 x)) + 2 \cdot 0}{2 y \cos^{3} (3 x)}$

Step 4.3.2.4

Factor $2$ out of $2 (- 9 \cdot - 1 y \cos^{2} (3 x) \sin (3 x)) + 2 (0)$ .

$\frac{2 (- 9 \cdot - 1 y \cos^{2} (3 x) \sin (3 x) + 0)}{2 y \cos^{3} (3 x)}$

Step 4.3.2.5

Cancel the common factors.

Tap for more steps...

Step 4.3.2.5.1

Factor $2$ out of $2 y \cos^{3} (3 x)$ .

$\frac{2 (- 9 \cdot - 1 y \cos^{2} (3 x) \sin (3 x) + 0)}{2 (y \cos^{3} (3 x))}$

Step 4.3.2.5.2

Cancel the common factor.

$\frac{2 (- 9 \cdot - 1 y \cos^{2} (3 x) \sin (3 x) + 0)}{2 (y \cos^{3} (3 x))}$

Step 4.3.2.5.3

Rewrite the expression.

$\frac{- 9 \cdot - 1 y \cos^{2} (3 x) \sin (3 x) + 0}{y \cos^{3} (3 x)}$

Step 4.3.3

Simplify the numerator.

Tap for more steps...

Step 4.3.3.1

Multiply $- 9$ by $- 1$ .

$\frac{9 y \cos^{2} (3 x) \sin (3 x) + 0}{y \cos^{3} (3 x)}$

Step 4.3.3.2

Add $9 y \cos^{2} (3 x) \sin (3 x)$ and $0$ .

$\frac{9 y \cos^{2} (3 x) \sin (3 x)}{y \cos^{3} (3 x)}$

Step 4.3.4

Cancel the common factor of $y$ .

Tap for more steps...

Step 4.3.4.1

Cancel the common factor.

$\frac{9 y \cos^{2} (3 x) \sin (3 x)}{y \cos^{3} (3 x)}$

Step 4.3.4.2

Rewrite the expression.

$\frac{9 \cos^{2} (3 x) \sin (3 x)}{\cos^{3} (3 x)}$

Step 4.3.5

Cancel the common factor of $\cos^{2} (3 x)$ and $\cos^{3} (3 x)$ .

Tap for more steps...

Step 4.3.5.1

Factor $\cos^{2} (3 x)$ out of $9 \cos^{2} (3 x) \sin (3 x)$ .

$\frac{\cos^{2} (3 x) (9 \sin (3 x))}{\cos^{3} (3 x)}$

Step 4.3.5.2

Cancel the common factors.

Tap for more steps...

Step 4.3.5.2.1

Factor $\cos^{2} (3 x)$ out of $\cos^{3} (3 x)$ .

$\frac{\cos^{2} (3 x) (9 \sin (3 x))}{\cos^{2} (3 x) \cos (3 x)}$

Step 4.3.5.2.2

Cancel the common factor.

$\frac{\cos^{2} (3 x) (9 \sin (3 x))}{\cos^{2} (3 x) \cos (3 x)}$

Step 4.3.5.2.3

Rewrite the expression.

$\frac{9 \sin (3 x)}{\cos (3 x)}$

Step 4.3.6

Separate fractions.

$\frac{9}{1} \cdot \frac{\sin (3 x)}{\cos (3 x)}$

Step 4.3.7

Convert from $\frac{\sin (3 x)}{\cos (3 x)}$ to $\tan (3 x)$ .

$\frac{9}{1} \tan (3 x)$

Step 4.3.8

Divide $9$ by $1$ .

$9 \tan (3 x)$

Step 4.4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} d x}$ .

$μ (x, y) = e^{\int 9 \tan (3 x) d x}$

Step 5

Evaluate the integral $e^{\int 9 \tan (3 x) d x}$ .

Tap for more steps...

Step 5.1

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$μ (x, y) = e^{9 \int \tan (3 x) d x}$

Step 5.2

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 5.2.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 5.2.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 5.2.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 5.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 \cdot 1$

Step 5.2.1.4

Multiply $3$ by $1$ .

$3$

Step 5.2.2

Rewrite the problem using $u$ and $d u$ .

$μ (x, y) = e^{9 \int \tan (u) \frac{1}{3} d u}$

Step 5.3

Combine $\tan (u)$ and $\frac{1}{3}$ .

$μ (x, y) = e^{9 \int \frac{\tan (u)}{3} d u}$

Step 5.4

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$μ (x, y) = e^{9 (\frac{1}{3} \int \tan (u) d u)}$

Step 5.5

Simplify.

Tap for more steps...

Step 5.5.1

Combine $\frac{1}{3}$ and $9$ .

$μ (x, y) = e^{\frac{9}{3} \int \tan (u) d u}$

Step 5.5.2

Cancel the common factor of $9$ and $3$ .

Tap for more steps...

Step 5.5.2.1

Factor $3$ out of $9$ .

$μ (x, y) = e^{\frac{3 \cdot 3}{3} \int \tan (u) d u}$

Step 5.5.2.2

Cancel the common factors.

Tap for more steps...

Step 5.5.2.2.1

Factor $3$ out of $3$ .

$μ (x, y) = e^{\frac{3 \cdot 3}{3 (1)} \int \tan (u) d u}$

Step 5.5.2.2.2

Cancel the common factor.

$μ (x, y) = e^{\frac{3 \cdot 3}{3 \cdot 1} \int \tan (u) d u}$

Step 5.5.2.2.3

Rewrite the expression.

$μ (x, y) = e^{\frac{3}{1} \int \tan (u) d u}$

Step 5.5.2.2.4

Divide $3$ by $1$ .

$μ (x, y) = e^{3 \int \tan (u) d u}$

Step 5.6

The integral of $\tan (u)$ with respect to $u$ is $\ln (| \sec (u) |)$ .

$μ (x, y) = e^{3 (\ln (| \sec (u) |) + C)}$

Step 5.7

Simplify.

$μ (x, y) = e^{3 \ln (| \sec (u) |)} + C$

Step 5.8

Replace all occurrences of $u$ with $3 x$ .

$μ (x, y) = e^{3 \ln (| \sec (3 x) |)} + C$

Step 5.9

Simplify each term.

Tap for more steps...

Step 5.9.1

Simplify $3 \ln (| \sec (3 x) |)$ by moving $3$ inside the logarithm.

$μ (x, y) = e^{\ln ({| \sec (3 x) |}^{3})} + C$

Step 5.9.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| \sec (3 x) |}^{3} + C$

Step 6

Multiply both sides of $\sin (3 x) d x + 2 y \cos^{3} (3 x) d y = 0$ by the integration factor $\sec^{3} (3 x)$ .

Tap for more steps...

Step 6.1

Multiply $\sin (3 x)$ by $\sec^{3} (3 x)$ .

$\sin (3 x) \sec^{3} (3 x) d x + 2 y \cos^{3} (3 x) d y = 0$

Step 6.2

Rewrite $\sec (3 x)$ in terms of sines and cosines.

$\sin (3 x) {(\frac{1}{\cos (3 x)})}^{3} d x + 2 y \cos^{3} (3 x) d y = 0$

Step 6.3

Apply the product rule to $\frac{1}{\cos (3 x)}$ .

$\sin (3 x) \frac{1^{3}}{\cos^{3} (3 x)} d x + 2 y \cos^{3} (3 x) d y = 0$

Step 6.4

One to any power is one.

$\sin (3 x) \frac{1}{\cos^{3} (3 x)} d x + 2 y \cos^{3} (3 x) d y = 0$

Step 6.5

Combine $\sin (3 x)$ and $\frac{1}{\cos^{3} (3 x)}$ .

$\frac{\sin (3 x)}{\cos^{3} (3 x)} d x + 2 y \cos^{3} (3 x) d y = 0$

Step 6.6

Factor $\cos (3 x)$ out of $\cos^{3} (3 x)$ .

$\frac{\sin (3 x)}{\cos (3 x) \cos^{2} (3 x)} d x + 2 y \cos^{3} (3 x) d y = 0$

Step 6.7

Separate fractions.

$\frac{1}{\cos^{2} (3 x)} \cdot \frac{\sin (3 x)}{\cos (3 x)} d x + 2 y \cos^{3} (3 x) d y = 0$

Step 6.8

Convert from $\frac{\sin (3 x)}{\cos (3 x)}$ to $\tan (3 x)$ .

$\frac{1}{\cos^{2} (3 x)} \tan (3 x) d x + 2 y \cos^{3} (3 x) d y = 0$

Step 6.9

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2}}{\cos^{2} (3 x)} \tan (3 x) d x + 2 y \cos^{3} (3 x) d y = 0$

Step 6.10

Rewrite $\frac{1^{2}}{\cos^{2} (3 x)}$ as ${(\frac{1}{\cos (3 x)})}^{2}$ .

${(\frac{1}{\cos (3 x)})}^{2} \tan (3 x) d x + 2 y \cos^{3} (3 x) d y = 0$

Step 6.11

Convert from $\frac{1}{\cos (3 x)}$ to $\sec (3 x)$ .

$\sec^{2} (3 x) \tan (3 x) d x + 2 y \cos^{3} (3 x) d y = 0$

Step 6.12

Multiply $2 y \cos^{3} (3 x)$ by $\sec^{3} (3 x)$ .

$\sec^{2} (3 x) \tan (3 x) d x + 2 y \cos^{3} (3 x) \sec^{3} (3 x) d y = 0$

Step 6.13

Rewrite $\sec (3 x)$ in terms of sines and cosines.

$\sec^{2} (3 x) \tan (3 x) d x + 2 y \cos^{3} (3 x) {(\frac{1}{\cos (3 x)})}^{3} d y = 0$

Step 6.14

Apply the product rule to $\frac{1}{\cos (3 x)}$ .

$\sec^{2} (3 x) \tan (3 x) d x + 2 y \cos^{3} (3 x) \frac{1^{3}}{\cos^{3} (3 x)} d y = 0$

Step 6.15

Cancel the common factor of $\cos^{3} (3 x)$ .

Tap for more steps...

Step 6.15.1

Factor $\cos^{3} (3 x)$ out of $2 y \cos^{3} (3 x)$ .

$\sec^{2} (3 x) \tan (3 x) d x + \cos^{3} (3 x) (2 y) \frac{1^{3}}{\cos^{3} (3 x)} d y = 0$

Step 6.15.2

Cancel the common factor.

$\sec^{2} (3 x) \tan (3 x) d x + \cos^{3} (3 x) (2 y) \frac{1^{3}}{\cos^{3} (3 x)} d y = 0$

Step 6.15.3

Rewrite the expression.

$\sec^{2} (3 x) \tan (3 x) d x + 2 y \cdot 1^{3} d y = 0$

Step 6.16

One to any power is one.

$\sec^{2} (3 x) \tan (3 x) d x + 2 y \cdot 1 d y = 0$

Step 6.17

Multiply $2$ by $1$ .

$\sec^{2} (3 x) \tan (3 x) d x + 2 y d y = 0$

Step 7

Set $f (x, y)$ equal to the integral of $N (x, y)$ .

$f (x, y) = \int 2 y d y$

Step 8

Integrate $N (x, y) = 2 y$ to find $f (x, y)$ .

Tap for more steps...

Step 8.1

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

$f (x, y) = 2 \int y d y$

Step 8.2

By the Power Rule, the integral of $y$ with respect to $y$ is $\frac{1}{2} y^{2}$ .

$f (x, y) = 2 (\frac{1}{2} y^{2} + C)$

Step 8.3

Simplify the answer.

Tap for more steps...

Step 8.3.1

Rewrite $2 (\frac{1}{2} y^{2} + C)$ as $2 (\frac{1}{2}) y^{2} + C$ .

$f (x, y) = 2 (\frac{1}{2}) y^{2} + C$

Step 8.3.2

Simplify.

Tap for more steps...

Step 8.3.2.1

Combine $2$ and $\frac{1}{2}$ .

$f (x, y) = \frac{2}{2} y^{2} + C$

Step 8.3.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.3.2.2.1

Cancel the common factor.

$f (x, y) = \frac{2}{2} y^{2} + C$

Step 8.3.2.2.2

Rewrite the expression.

$f (x, y) = 1 y^{2} + C$

Step 8.3.2.3

Multiply $y^{2}$ by $1$ .

$f (x, y) = y^{2} + C$

Step 9

Since the integral of $g (x)$ will contain an integration constant, we can replace $C$ with $g (x)$ .

$f (x, y) = y^{2} + g (x)$

Step 10

Set $\frac{\partial f}{\partial x} = M (x, y)$ .

$\frac{\partial f}{\partial x} = \sec^{2} (3 x) \tan (3 x)$

Step 11

Find $\frac{\partial f}{\partial x}$ .

Tap for more steps...

Step 11.1

Differentiate $f$ with respect to $x$ .

$\frac{d}{d x} [y^{2} + g (x)] = \sec^{2} (3 x) \tan (3 x)$

Step 11.2

By the Sum Rule, the derivative of $y^{2} + g (x)$ with respect to $x$ is $\frac{d}{d x} [y^{2}] + \frac{d}{d x} [g (x)]$ .

$\frac{d}{d x} [y^{2}] + \frac{d}{d x} [g (x)] = \sec^{2} (3 x) \tan (3 x)$

Step 11.3

Since $y^{2}$ is constant with respect to $x$ , the derivative of $y^{2}$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [g (x)] = \sec^{2} (3 x) \tan (3 x)$

Step 11.4

Differentiate using the function rule which states that the derivative of $g (x)$ is $\frac{d g}{d x}$ .

$0 + \frac{d g}{d x} = \sec^{2} (3 x) \tan (3 x)$

Step 11.5

Add $0$ and $\frac{d g}{d x}$ .

$\frac{d g}{d x} = \sec^{2} (3 x) \tan (3 x)$

Step 12

Find the antiderivative of $\sec^{2} (3 x) \tan (3 x)$ to find $g (x)$ .

Tap for more steps...

Step 12.1

Integrate both sides of $\frac{d g}{d x} = \sec^{2} (3 x) \tan (3 x)$ .

$\int \frac{d g}{d x} d x = \int \sec^{2} (3 x) \tan (3 x) d x$

Step 12.2

Evaluate $\int \frac{d g}{d x} d x$ .

$g (x) = \int \sec^{2} (3 x) \tan (3 x) d x$

Step 12.3

Let $u_{2} = \sec (3 x)$ . Then $d u_{2} = 3 \sec (3 x) \tan (3 x) d x$ , so $\frac{1}{3} d u_{2} = \sec (3 x) \tan (3 x) d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 12.3.1

Let $u_{2} = \sec (3 x)$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 12.3.1.1

Differentiate $\sec (3 x)$ .

$\frac{d}{d x} [\sec (3 x)]$

Step 12.3.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = 3 x$ .

Tap for more steps...

Step 12.3.1.2.1

To apply the Chain Rule, set $u_{1}$ as $3 x$ .

$\frac{d}{d u_{1}} [\sec (u_{1})] \frac{d}{d x} [3 x]$

Step 12.3.1.2.2

The derivative of $\sec (u_{1})$ with respect to $u_{1}$ is $\sec (u_{1}) \tan (u_{1})$ .

$\sec (u_{1}) \tan (u_{1}) \frac{d}{d x} [3 x]$

Step 12.3.1.2.3

Replace all occurrences of $u_{1}$ with $3 x$ .

$\sec (3 x) \tan (3 x) \frac{d}{d x} [3 x]$

Step 12.3.1.3

Differentiate.

Tap for more steps...

Step 12.3.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\sec (3 x) \tan (3 x) (3 \frac{d}{d x} [x])$

Step 12.3.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\sec (3 x) \tan (3 x) (3 \cdot 1)$

Step 12.3.1.3.3

Simplify the expression.

Tap for more steps...

Step 12.3.1.3.3.1

Multiply $3$ by $1$ .

$\sec (3 x) \tan (3 x) \cdot 3$

Step 12.3.1.3.3.2

Move $3$ to the left of $\sec (3 x) \tan (3 x)$ .

$3 \sec (3 x) \tan (3 x)$

Step 12.3.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$g (x) = \int u_{2} \frac{1}{3} d u_{2}$

Step 12.4

Combine $u_{2}$ and $\frac{1}{3}$ .

$g (x) = \int \frac{u_{2}}{3} d u_{2}$

Step 12.5

Since $\frac{1}{3}$ is constant with respect to $u_{2}$ , move $\frac{1}{3}$ out of the integral.

$g (x) = \frac{1}{3} \int u_{2} d u_{2}$

Step 12.6

By the Power Rule, the integral of $u_{2}$ with respect to $u_{2}$ is $\frac{1}{2} {u_{2}}^{2}$ .

$g (x) = \frac{1}{3} (\frac{1}{2} {u_{2}}^{2} + C)$

Step 12.7

Rewrite $\frac{1}{3} (\frac{1}{2} {u_{2}}^{2} + C)$ as $\frac{1}{3} \cdot \frac{1}{2} {u_{2}}^{2} + C$ .

$g (x) = \frac{1}{3} \cdot \frac{1}{2} {u_{2}}^{2} + C$

Step 12.8

Simplify.

Tap for more steps...

Step 12.8.1

Multiply $\frac{1}{3}$ by $\frac{1}{2}$ .

$g (x) = \frac{1}{3 \cdot 2} {u_{2}}^{2} + C$

Step 12.8.2

Multiply $3$ by $2$ .

$g (x) = \frac{1}{6} {u_{2}}^{2} + C$

Step 12.9

Replace all occurrences of $u_{2}$ with $\sec (3 x)$ .

$g (x) = \frac{1}{6} \sec^{2} (3 x) + C$

Step 13

Substitute for $g (x)$ in $f (x, y) = y^{2} + g (x)$ .

$f (x, y) = y^{2} + \frac{1}{6} \sec^{2} (3 x) + C$

Step 14

Combine $\frac{1}{6}$ and $\sec^{2} (3 x)$ .

$f (x, y) = y^{2} + \frac{\sec^{2} (3 x)}{6} + C$