Calculus Examples

\int \tan^{2} (x) d x

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

Step 1

Using the Pythagorean Identity, rewrite

\tan^{2} (x)

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

- 1 + \sec^{2} (x)

$- 1 + \sec^{2} (x)$ .

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

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

Step 2

Split the single integral into multiple integrals.

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

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

Step 3

Apply the constant rule.

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

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

Step 4

Since the derivative of

\tan (x)

$\tan (x)$ is

\sec^{2} (x)

$\sec^{2} (x)$ , the integral of

\sec^{2} (x)

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

\tan (x)

$\tan (x)$ .

- x + C + \tan (x) + C

$- x + C + \tan (x) + C$

Step 5

Simplify.

- x + \tan (x) + C

$- x + \tan (x) + C$

\int_{}^{} \tan^{2} x

$\int_{}^{} \tan^{2} x$

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