Calculus Examples

$\int_{0}^{1} \frac{2 \arctan (x)}{1 + x^{2}} d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{0}^{1} \frac{\arctan (x)}{1 + x^{2}} d x$

Step 2

Let $u = \arctan (x)$ . Then $d u = \frac{1}{1 + x^{2}} d x$ , so $- d u = \frac{- 1}{x^{2} + 1} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = \arctan (x)$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $\arctan (x)$ .

$\frac{d}{d x} [\arctan (x)]$

Step 2.1.2

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

$\frac{1}{1 + x^{2}}$

Step 2.1.3

Reorder terms.

$\frac{1}{x^{2} + 1}$

Step 2.1.4

Rearrange terms.

$\frac{1}{1 + x^{2}}$

Step 2.2

Substitute the lower limit in for $x$ in $u = \arctan (x)$ .

$u_{lower} = \arctan (0)$

Step 2.3

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

$u_{lower} = 0$

Step 2.4

Substitute the upper limit in for $x$ in $u = \arctan (x)$ .

$u_{upper} = \arctan (1)$

Step 2.5

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

$u_{upper} = \frac{π}{4}$

Step 2.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = \frac{π}{4}$

Step 2.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$2 \int_{0}^{\frac{π}{4}} u d u$

Step 3

By the Power Rule, the integral of $u$ with respect to $u$ is $\frac{1}{2} u^{2}$ .

$2 (\frac{1}{2} u^{2}]_{0}^{\frac{π}{4}})$

Step 4

Combine $\frac{1}{2}$ and $u^{2}$ .

$2 (\frac{u^{2}}{2}]_{0}^{\frac{π}{4}})$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{u^{2}}{2}$ at $\frac{π}{4}$ and at $0$ .

$2 ((\frac{{(\frac{π}{4})}^{2}}{2}) - \frac{0^{2}}{2})$

Step 5.2

Simplify.

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Step 5.2.1

Raising $0$ to any positive power yields $0$ .

$2 (\frac{{(\frac{π}{4})}^{2}}{2} - \frac{0}{2})$

Step 5.2.2

Cancel the common factor of $0$ and $2$ .

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Step 5.2.2.1

Factor $2$ out of $0$ .

$2 (\frac{{(\frac{π}{4})}^{2}}{2} - \frac{2 (0)}{2})$

Step 5.2.2.2

Cancel the common factors.

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Step 5.2.2.2.1

Factor $2$ out of $2$ .

$2 (\frac{{(\frac{π}{4})}^{2}}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 5.2.2.2.2

Cancel the common factor.

$2 (\frac{{(\frac{π}{4})}^{2}}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 5.2.2.2.3

Rewrite the expression.

$2 (\frac{{(\frac{π}{4})}^{2}}{2} - \frac{0}{1})$

Step 5.2.2.2.4

Divide $0$ by $1$ .

$2 (\frac{{(\frac{π}{4})}^{2}}{2} - 0)$

Step 5.2.3

Multiply $- 1$ by $0$ .

$2 (\frac{{(\frac{π}{4})}^{2}}{2} + 0)$

Step 5.2.4

Add $\frac{{(\frac{π}{4})}^{2}}{2}$ and $0$ .

$2 \frac{{(\frac{π}{4})}^{2}}{2}$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $2$ .

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Step 6.1.1

Cancel the common factor.

$2 \frac{{(\frac{π}{4})}^{2}}{2}$

Step 6.1.2

Rewrite the expression.

${(\frac{π}{4})}^{2}$

Step 6.2

Apply the product rule to $\frac{π}{4}$ .

$\frac{π^{2}}{4^{2}}$

Step 6.3

Raise $4$ to the power of $2$ .

$\frac{π^{2}}{16}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{π^{2}}{16}$

Decimal Form:

$0.61685027 \dots$