Calculus Examples

$\int_{0}^{1} \frac{5}{x^{2} + 1} d x$

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int_{0}^{1} \frac{1}{x^{2} + 1} d x$

Step 3

Simplify the expression.

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Step 3.1

Reorder $x^{2}$ and $1$ .

$5 \int_{0}^{1} \frac{1}{1 + x^{2}} d x$

Step 3.2

Rewrite $1$ as $1^{2}$ .

$5 \int_{0}^{1} \frac{1}{1^{2} + x^{2}} d x$

Step 4

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x)]_{0}^{1}$ .

$5 (\arctan (x)]_{0}^{1})$

Step 5

Evaluate $\arctan (x)$ at $1$ and at $0$ .

$5 (\arctan (1) - \arctan (0))$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

$5 (\frac{π}{4} - \arctan (0))$

Step 6.1.2

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

$5 (\frac{π}{4} - 0)$

Step 6.1.3

Multiply $- 1$ by $0$ .

$5 (\frac{π}{4} + 0)$

Step 6.2

Add $\frac{π}{4}$ and $0$ .

$5 \frac{π}{4}$

Step 6.3

Combine $5$ and $\frac{π}{4}$ .

$\frac{5 π}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{5 π}{4}$

Decimal Form:

$3.92699081 \dots$

Step 8