Calculus Examples

$\int_{0}^{\frac{π}{4}} \frac{1}{\cos^{2} (x)} d x$

Step 1

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

$\int_{0}^{\frac{π}{4}} \sec^{2} (x) d x$

Step 2

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

$\tan (x)]_{0}^{\frac{π}{4}}$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\tan (x)$ at $\frac{π}{4}$ and at $0$ .

$\tan (\frac{π}{4}) - \tan (0)$

Step 3.2

Simplify.

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Step 3.2.1

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

$1 - \tan (0)$

Step 3.2.2

The exact value of $\tan (0)$ is $0$ .

$1 - 0$

Step 3.2.3

Multiply $- 1$ by $0$ .

$1 + 0$

Step 3.2.4

Add $1$ and $0$ .

$1$