Calculus Examples

$\int 2 x^{3} \sin (3 x^{4}) d x$

Step 1

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

$2 \int x^{3} \sin (3 x^{4}) d x$

Step 2

Let $u_{1} = x^{2}$ . Then $d u_{1} = 2 x d x$ , so $\frac{1}{2} d u_{1} = x d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 2.1

Let $u_{1} = x^{2}$ . Find $\frac{d u_{1}}{d x}$ .

Tap for more steps...

Step 2.1.1

Differentiate $x^{2}$ .

$\frac{d}{d x} [x^{2}]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x$

Step 2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$2 \int u_{1} \sin (3 {\sqrt{u_{1}}}^{4}) \frac{1}{2} d u_{1}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Rewrite ${\sqrt{u_{1}}}^{4}$ as ${u_{1}}^{2}$ .

Tap for more steps...

Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u_{1}}$ as ${u_{1}}^{\frac{1}{2}}$ .

$2 \int u_{1} \sin (3 {({u_{1}}^{\frac{1}{2}})}^{4}) \frac{1}{2} d u_{1}$

Step 3.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 \int u_{1} \sin (3 {u_{1}}^{\frac{1}{2} \cdot 4}) \frac{1}{2} d u_{1}$

Step 3.1.3

Combine $\frac{1}{2}$ and $4$ .

$2 \int u_{1} \sin (3 {u_{1}}^{\frac{4}{2}}) \frac{1}{2} d u_{1}$

Step 3.1.4

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 3.1.4.1

Factor $2$ out of $4$ .

$2 \int u_{1} \sin (3 {u_{1}}^{\frac{2 \cdot 2}{2}}) \frac{1}{2} d u_{1}$

Step 3.1.4.2

Cancel the common factors.

Tap for more steps...

Step 3.1.4.2.1

Factor $2$ out of $2$ .

$2 \int u_{1} \sin (3 {u_{1}}^{\frac{2 \cdot 2}{2 (1)}}) \frac{1}{2} d u_{1}$

Step 3.1.4.2.2

Cancel the common factor.

$2 \int u_{1} \sin (3 {u_{1}}^{\frac{2 \cdot 2}{2 \cdot 1}}) \frac{1}{2} d u_{1}$

Step 3.1.4.2.3

Rewrite the expression.

$2 \int u_{1} \sin (3 {u_{1}}^{\frac{2}{1}}) \frac{1}{2} d u_{1}$

Step 3.1.4.2.4

Divide $2$ by $1$ .

$2 \int u_{1} \sin (3 {u_{1}}^{2}) \frac{1}{2} d u_{1}$

Step 3.2

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

$2 \int \frac{u_{1}}{2} \sin (3 {u_{1}}^{2}) d u_{1}$

Step 3.3

Combine $\frac{u_{1}}{2}$ and $\sin (3 {u_{1}}^{2})$ .

$2 \int \frac{u_{1} \sin (3 {u_{1}}^{2})}{2} d u_{1}$

Step 4

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

$2 (\frac{1}{2} \int u_{1} \sin (3 {u_{1}}^{2}) d u_{1})$

Step 5

Simplify.

Tap for more steps...

Step 5.1

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

$\frac{2}{2} \int u_{1} \sin (3 {u_{1}}^{2}) d u_{1}$

Step 5.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.1

Cancel the common factor.

$\frac{2}{2} \int u_{1} \sin (3 {u_{1}}^{2}) d u_{1}$

Step 5.2.2

Rewrite the expression.

$1 \int u_{1} \sin (3 {u_{1}}^{2}) d u_{1}$

Step 5.3

Multiply $\int u_{1} \sin (3 {u_{1}}^{2}) d u_{1}$ by $1$ .

$\int u_{1} \sin (3 {u_{1}}^{2}) d u_{1}$

Step 6

Let $u_{2} = 3 {u_{1}}^{2}$ . Then $d u_{2} = 6 u_{1} d u_{1}$ , so $\frac{1}{6} d u_{2} = u_{1} d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 6.1

Let $u_{2} = 3 {u_{1}}^{2}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

Tap for more steps...

Step 6.1.1

Differentiate $3 {u_{1}}^{2}$ .

$\frac{d}{d u_{1}} [3 {u_{1}}^{2}]$

Step 6.1.2

Since $3$ is constant with respect to $u_{1}$ , the derivative of $3 {u_{1}}^{2}$ with respect to $u_{1}$ is $3 \frac{d}{d u_{1}} [{u_{1}}^{2}]$ .

$3 \frac{d}{d u_{1}} [{u_{1}}^{2}]$

Step 6.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$3 (2 u_{1})$

Step 6.1.4

Multiply $2$ by $3$ .

$6 u_{1}$

Step 6.2

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

$\int \sin (u_{2}) \frac{1}{6} d u_{2}$

Step 7

Combine $\sin (u_{2})$ and $\frac{1}{6}$ .

$\int \frac{\sin (u_{2})}{6} d u_{2}$

Step 8

Since $\frac{1}{6}$ is constant with respect to $u_{2}$ , move $\frac{1}{6}$ out of the integral.

$\frac{1}{6} \int \sin (u_{2}) d u_{2}$

Step 9

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

$\frac{1}{6} (- \cos (u_{2}) + C)$

Step 10

Simplify.

Tap for more steps...

Step 10.1

Simplify.

$\frac{1}{6} (- \cos (u_{2})) + C$

Step 10.2

Combine $\frac{1}{6}$ and $\cos (u_{2})$ .

$- \frac{\cos (u_{2})}{6} + C$

Step 11

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 11.1

Replace all occurrences of $u_{2}$ with $3 {u_{1}}^{2}$ .

$- \frac{\cos (3 {u_{1}}^{2})}{6} + C$

Step 11.2

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$- \frac{\cos (3 {(x^{2})}^{2})}{6} + C$

Step 12

Simplify.

Tap for more steps...

Step 12.1

Multiply the exponents in ${(x^{2})}^{2}$ .

Tap for more steps...

Step 12.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{\cos (3 x^{2 \cdot 2})}{6} + C$

Step 12.1.2

Multiply $2$ by $2$ .

$- \frac{\cos (3 x^{4})}{6} + C$

Step 12.2

Reorder terms.

$- \frac{1}{6} \cos (3 x^{4}) + C$