Calculus Examples

$\tan^{2} (θ) + 1$

Step 1

Write $\tan^{2} (θ) + 1$ as a function.

$f (θ) = \tan^{2} (θ) + 1$

Step 2

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

$F (θ) = \int f (θ) d θ$

Step 3

Set up the integral to solve.

$F (θ) = \int \tan^{2} (θ) + 1 d θ$

Step 4

Split the single integral into multiple integrals.

$\int \tan^{2} (θ) d θ + \int d θ$

Step 5

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

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

Apply the constant rule.

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

Step 8

Since the derivative of $\tan (θ)$ is $\sec^{2} (θ)$ , the integral of $\sec^{2} (θ)$ is $\tan (θ)$ .

$- θ + C + \tan (θ) + C + \int d θ$

Step 9

Apply the constant rule.

$- θ + C + \tan (θ) + C + θ + C$

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

Add $- θ$ and $θ$ .

$C + \tan (θ) + C + 0 + C$

Step 10.1.2

Add $C + \tan (θ) + C$ and $0$ .

$C + \tan (θ) + C + C$

Step 10.2

Simplify.

$\tan (θ) + C$

Step 11

The answer is the antiderivative of the function $f (θ) = \tan^{2} (θ) + 1$ .

$F (θ) =$ $\tan (θ) + C$