Calculus Examples

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

Step 1

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

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Step 1.1

Let $u = x^{2} + 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{2} + 1$ .

$\frac{d}{d x} [x^{2} + 1]$

Step 1.1.2

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [1]$

Step 1.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 x + 0$

Step 1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = x^{2} + 1$ .

$u_{lower} = 0^{2} + 1$

Step 1.3

Simplify.

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Step 1.3.1

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

$u_{lower} = 0 + 1$

Step 1.3.2

Add $0$ and $1$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = x^{2} + 1$ .

$u_{upper} = 1^{2} + 1$

Step 1.5

Simplify.

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Step 1.5.1

One to any power is one.

$u_{upper} = 1 + 1$

Step 1.5.2

Add $1$ and $1$ .

$u_{upper} = 2$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 2$

Step 1.7

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

$\int_{1}^{2} u^{2} d u$

Step 2

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$\frac{1}{3} u^{3}]_{1}^{2}$

Step 3

Combine fractions.

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Step 3.1

Evaluate $\frac{1}{3} u^{3}$ at $2$ and at $1$ .

$(\frac{1}{3} \cdot 2^{3}) - \frac{1}{3} \cdot 1^{3}$

Step 3.2

Raise $2$ to the power of $3$ .

$\frac{1}{3} \cdot 8 - \frac{1}{3} \cdot 1^{3}$

Step 3.3

Combine $\frac{1}{3}$ and $8$ .

$\frac{8}{3} - \frac{1}{3} \cdot 1^{3}$

Step 3.4

Simplify the expression.

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Step 3.4.1

One to any power is one.

$\frac{8}{3} - \frac{1}{3} \cdot 1$

Step 3.4.2

Multiply $- 1$ by $1$ .

$\frac{8}{3} - \frac{1}{3}$

Step 3.4.3

Simplify.

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Step 3.4.3.1

Combine the numerators over the common denominator.

$\frac{8 - 1}{3}$

Step 3.4.3.2

Subtract $1$ from $8$ .

$\frac{7}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{3}$

Decimal Form:

$2.\overline{3}$

Mixed Number Form:

$2 \frac{1}{3}$