Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{\sec^{2} (y)}$ .

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

Step 1.2

Cancel the common factor of $\sec^{2} (y)$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sec^{2} (y)} d y = \int \frac{1}{1 + x^{2}} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify.

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Step 2.2.1.1

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

$\int \frac{1^{2}}{\sec^{2} (y)} d y = \int \frac{1}{1 + x^{2}} d x$

Step 2.2.1.2

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

$\int {(\frac{1}{\sec (y)})}^{2} d y = \int \frac{1}{1 + x^{2}} d x$

Step 2.2.1.3

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

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

Step 2.2.1.4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (y)}$ .

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

Step 2.2.1.5

Multiply $\cos (y)$ by $1$ .

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

Step 2.2.2

Use the half-angle formula to rewrite $\cos^{2} (y)$ as $\frac{1 + \cos (2 y)}{2}$ .

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

Step 2.2.3

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

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

Step 2.2.4

Split the single integral into multiple integrals.

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

Step 2.2.5

Apply the constant rule.

$\frac{1}{2} (y + C_{1} + \int \cos (2 y) d y) = \int \frac{1}{1 + x^{2}} d x$

Step 2.2.6

Let $u = 2 y$ . Then $d u = 2 d y$ , so $\frac{1}{2} d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.6.1

Let $u = 2 y$ . Find $\frac{d u}{d y}$ .

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Step 2.2.6.1.1

Differentiate $2 y$ .

$\frac{d}{d y} [2 y]$

Step 2.2.6.1.2

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

$2 \frac{d}{d y} [y]$

Step 2.2.6.1.3

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

$2 \cdot 1$

Step 2.2.6.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2.6.2

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

$\frac{1}{2} (y + C_{1} + \int \cos (u) \frac{1}{2} d u) = \int \frac{1}{1 + x^{2}} d x$

Step 2.2.7

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

$\frac{1}{2} (y + C_{1} + \int \frac{\cos (u)}{2} d u) = \int \frac{1}{1 + x^{2}} d x$

Step 2.2.8

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

$\frac{1}{2} (y + C_{1} + \frac{1}{2} \int \cos (u) d u) = \int \frac{1}{1 + x^{2}} d x$

Step 2.2.9

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\frac{1}{2} (y + C_{1} + \frac{1}{2} (\sin (u) + C_{2})) = \int \frac{1}{1 + x^{2}} d x$

Step 2.2.10

Simplify.

$\frac{1}{2} (y + \frac{1}{2} \sin (u)) + C_{3} = \int \frac{1}{1 + x^{2}} d x$

Step 2.2.11

Replace all occurrences of $u$ with $2 y$ .

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

Step 2.2.12

Simplify.

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Step 2.2.12.1

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

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

Step 2.2.12.2

Apply the distributive property.

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

Step 2.2.12.3

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

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

Step 2.2.12.4

Multiply $\frac{1}{2} \cdot \frac{\sin (2 y)}{2}$ .

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Step 2.2.12.4.1

Multiply $\frac{1}{2}$ by $\frac{\sin (2 y)}{2}$ .

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

Step 2.2.12.4.2

Multiply $2$ by $2$ .

$\frac{y}{2} + \frac{\sin (2 y)}{4} + C_{3} = \int \frac{1}{1 + x^{2}} d x$

Step 2.2.13

Reorder terms.

$\frac{1}{2} y + \frac{1}{4} \sin (2 y) + C_{3} = \int \frac{1}{1 + x^{2}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$\frac{1}{2} y + \frac{1}{4} \sin (2 y) + C_{3} = \int \frac{1}{1^{2} + x^{2}} d x$

Step 2.3.2

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x) + C_{4}$ .

$\frac{1}{2} y + \frac{1}{4} \sin (2 y) + C_{3} = \arctan (x) + C_{4}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$\frac{1}{2} y + \frac{1}{4} \sin (2 y) = \arctan (x) + K$