Calculus Examples

$\sin (x) \tan^{2} (x)$

Step 1

Write $\sin (x) \tan^{2} (x)$ as a function.

$f (x) = \sin (x) \tan^{2} (x)$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \sin (x) \tan^{2} (x) d x$

Step 4

Using the Pythagorean Identity, rewrite $\tan^{2} (x)$ as $- 1 + \sec^{2} (x)$ .

$\int \sin (x) (- 1 + \sec^{2} (x)) d x$

Step 5

Simplify terms.

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Step 5.1

Apply the distributive property.

$\int \sin (x) \cdot - 1 + \sin (x) \sec^{2} (x) d x$

Step 5.2

Simplify each term.

$\int - \sin (x) + \sin (x) \sec^{2} (x) d x$

Step 6

Split the single integral into multiple integrals.

$\int - \sin (x) d x + \int \sin (x) \sec^{2} (x) d x$

Step 7

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

$- \int \sin (x) d x + \int \sin (x) \sec^{2} (x) d x$

Step 8

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

$- (- \cos (x) + C) + \int \sin (x) \sec^{2} (x) d x$

Step 9

Simplify $\sin (x) \sec^{2} (x)$ .

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Step 9.1

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

$- (- \cos (x) + C) + \int \sin (x) {(\frac{1}{\cos (x)})}^{2} d x$

Step 9.2

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

$- (- \cos (x) + C) + \int \sin (x) \frac{1^{2}}{\cos^{2} (x)} d x$

Step 9.3

One to any power is one.

$- (- \cos (x) + C) + \int \sin (x) \frac{1}{\cos^{2} (x)} d x$

Step 9.4

Combine $\sin (x)$ and $\frac{1}{\cos^{2} (x)}$ .

$- (- \cos (x) + C) + \int \frac{\sin (x)}{\cos^{2} (x)} d x$

Step 9.5

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

$- (- \cos (x) + C) + \int \frac{\sin (x)}{\cos (x) \cos (x)} d x$

Step 9.6

Separate fractions.

$- (- \cos (x) + C) + \int \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} d x$

Step 9.7

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

$- (- \cos (x) + C) + \int \frac{1}{\cos (x)} \tan (x) d x$

Step 9.8

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

$- (- \cos (x) + C) + \int \sec (x) \tan (x) d x$

Step 10

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

$- (- \cos (x) + C) + \sec (x) + C$

Step 11

Simplify.

$\cos (x) + \sec (x) + C$

Step 12

The answer is the antiderivative of the function $f (x) = \sin (x) \tan^{2} (x)$ .

$F (x) =$ $\cos (x) + \sec (x) + C$