Calculus Examples

$\int_{- 1}^{1} 3 x^{2} \sqrt{x^{3} + 5} d x$

Step 1

Let $u = x^{3} + 5$ . Then $d u = 3 x^{2} d x$ , so $\frac{1}{3 x^{2}} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{3} + 5$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{3} + 5$ .

$\frac{d}{d x} [x^{3} + 5]$

Step 1.1.2

By the Sum Rule, the derivative of $x^{3} + 5$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [5]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [5]$

Step 1.1.4

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$3 x^{2} + 0$

Step 1.1.5

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 1.2

Substitute the lower limit in for $x$ in $u = x^{3} + 5$ .

$u_{lower} = {(- 1)}^{3} + 5$

Step 1.3

Simplify.

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Step 1.3.1

Raise $- 1$ to the power of $3$ .

$u_{lower} = - 1 + 5$

Step 1.3.2

Add $- 1$ and $5$ .

$u_{lower} = 4$

Step 1.4

Substitute the upper limit in for $x$ in $u = x^{3} + 5$ .

$u_{upper} = 1^{3} + 5$

Step 1.5

Simplify.

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Step 1.5.1

One to any power is one.

$u_{upper} = 1 + 5$

Step 1.5.2

Add $1$ and $5$ .

$u_{upper} = 6$

Step 1.6

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

$u_{lower} = 4$

$u_{upper} = 6$

Step 1.7

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

$\int_{4}^{6} \sqrt{u} d u$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\int_{4}^{6} u^{\frac{1}{2}} d u$

Step 3

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

$\frac{2}{3} u^{\frac{3}{2}}]_{4}^{6}$

Step 4

Combine fractions.

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Step 4.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $6$ and at $4$ .

$(\frac{2}{3} \cdot 6^{\frac{3}{2}}) - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 4.2

Combine $\frac{2}{3}$ and $6^{\frac{3}{2}}$ .

$\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 4.3

Simplify the expression.

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Step 4.3.1

Simplify.

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Step 4.3.1.1

Rewrite $4$ as $2^{2}$ .

$\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot {(2^{2})}^{\frac{3}{2}}$

Step 4.3.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 2^{2 (\frac{3}{2})}$

Step 4.3.1.3

Cancel the common factor of $2$ .

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Step 4.3.1.3.1

Cancel the common factor.

$\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 2^{2 (\frac{3}{2})}$

Step 4.3.1.3.2

Rewrite the expression.

$\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 2^{3}$

Step 4.3.1.4

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

$\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 8$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Multiply $8$ by $- 1$ .

$\frac{2 \cdot 6^{\frac{3}{2}}}{3} - 8 (\frac{2}{3})$

Step 4.3.2.2

Combine $- 8$ and $\frac{2}{3}$ .

$\frac{2 \cdot 6^{\frac{3}{2}}}{3} + \frac{- 8 \cdot 2}{3}$

Step 4.3.2.3

Multiply $- 8$ by $2$ .

$\frac{2 \cdot 6^{\frac{3}{2}}}{3} + \frac{- 16}{3}$

Step 4.3.2.4

Move the negative in front of the fraction.

$\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{16}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{16}{3}$

Decimal Form:

$4.46462563 \dots$