Calculus Examples

$\int_{- \frac{π}{4}}^{\frac{π}{4}} \sec^{2} (x) d x$

Step 1

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

$\tan (x)]_{- \frac{π}{4}}^{\frac{π}{4}}$

Step 2

Simplify the answer.

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Step 2.1

Evaluate $\tan (x)$ at $\frac{π}{4}$ and at $- \frac{π}{4}$ .

$\tan (\frac{π}{4}) - \tan (- \frac{π}{4})$

Step 2.2

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$1 - \tan (- \frac{π}{4})$

Step 2.3

Simplify.

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Step 2.3.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$1 - \tan (\frac{7 π}{4})$

Step 2.3.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the fourth quadrant.

$1 - - \tan (\frac{π}{4})$

Step 2.3.3

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$1 - (- 1 \cdot 1)$

Step 2.3.4

Multiply $- 1$ by $1$ .

$1 - - 1$

Step 2.3.5

Multiply $- 1$ by $- 1$ .

$1 + 1$

Step 2.3.6

Add $1$ and $1$ .

$2$