Calculus Examples

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

Step 1

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

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply the constant rule.

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

Step 4

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

$- x]_{0}^{\frac{π}{4}} + \tan (x)]_{0}^{\frac{π}{4}}$

Step 5

Simplify the answer.

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Step 5.1

Combine $- x]_{0}^{\frac{π}{4}}$ and $\tan (x)]_{0}^{\frac{π}{4}}$ .

$- x + \tan (x)]_{0}^{\frac{π}{4}}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

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

$(- \frac{π}{4} + \tan (\frac{π}{4})) - (- 0 + \tan (0))$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Multiply $- 1$ by $0$ .

$- \frac{π}{4} + \tan (\frac{π}{4}) - (0 + \tan (0))$

Step 5.2.2.2

Add $0$ and $\tan (0)$ .

$- \frac{π}{4} + \tan (\frac{π}{4}) - \tan (0)$

Step 5.3

Simplify.

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Step 5.3.1

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

$- \frac{π}{4} + 1 - \tan (0)$

Step 5.3.2

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

$- \frac{π}{4} + 1 - 0$

Step 5.3.3

Multiply $- 1$ by $0$ .

$- \frac{π}{4} + 1 + 0$

Step 5.3.4

Add $- \frac{π}{4} + 1$ and $0$ .

$- \frac{π}{4} + 1$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{π}{4} + 1$

Decimal Form:

$0.21460183 \dots$