Calculus Examples

$\frac{d y}{d x} = \frac{{(1 - \tan (x))}^{5}}{\cos^{2} (x)}$

Step 1

Rewrite the equation.

$d y = \frac{{(1 - \tan (x))}^{5}}{\cos^{2} (x)} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{{(1 - \tan (x))}^{5}}{\cos^{2} (x)} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{{(1 - \tan (x))}^{5}}{\cos^{2} (x)} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

$y + C_{1} = \int {(1 - \tan (x))}^{5} \sec^{2} (x) d x$

Step 2.3.2

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

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Step 2.3.2.1

Let $u = 1 - \tan (x)$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $1 - \tan (x)$ .

$\frac{d}{d x} [1 - \tan (x)]$

Step 2.3.2.1.2

Differentiate.

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Step 2.3.2.1.2.1

By the Sum Rule, the derivative of $1 - \tan (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \tan (x)]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- \tan (x)]$

Step 2.3.2.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \tan (x)]$

Step 2.3.2.1.3

Evaluate $\frac{d}{d x} [- \tan (x)]$ .

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Step 2.3.2.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \tan (x)$ with respect to $x$ is $- \frac{d}{d x} [\tan (x)]$ .

$0 - \frac{d}{d x} [\tan (x)]$

Step 2.3.2.1.3.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$0 - \sec^{2} (x)$

Step 2.3.2.1.4

Subtract $\sec^{2} (x)$ from $0$ .

$- \sec^{2} (x)$

Step 2.3.2.2

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

$y + C_{1} = \int - u^{5} d u$

Step 2.3.3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$y + C_{1} = - \int u^{5} d u$

Step 2.3.4

By the Power Rule, the integral of $u^{5}$ with respect to $u$ is $\frac{1}{6} u^{6}$ .

$y + C_{1} = - (\frac{1}{6} u^{6} + C_{2})$

Step 2.3.5

Rewrite $- (\frac{1}{6} u^{6} + C_{2})$ as $- \frac{1}{6} u^{6} + C_{2}$ .

$y + C_{1} = - \frac{1}{6} u^{6} + C_{2}$

Step 2.3.6

Replace all occurrences of $u$ with $1 - \tan (x)$ .

$y + C_{1} = - \frac{1}{6} {(1 - \tan (x))}^{6} + C_{2}$

Step 2.4

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

$y = - \frac{1}{6} {(1 - \tan (x))}^{6} + K$