Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Rewrite $\frac{\sec^{2} (x)}{\tan^{2} (x)}$ as ${(\frac{\sec (x)}{\tan (x)})}^{2}$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Cancel the common factor of $\cos (x)$ .

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Step 2.3.5.1

Cancel the common factor.

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

Step 2.3.5.2

Rewrite the expression.

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

Step 2.3.6

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

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

Step 2.3.7

Since the derivative of $- \cot (x)$ is $\csc^{2} (x)$ , the integral of $\csc^{2} (x)$ is $- \cot (x)$ .

$y + C_{1} = - \cot (x) + C_{2}$

Step 2.4

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

$y = - \cot (x) + K$