Calculus Examples

$\int_{0}^{1} 5 x^{4} {(3 + x^{5})}^{4} d x$

Step 1

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

$5 \int_{0}^{1} x^{4} {(3 + x^{5})}^{4} d x$

Step 2

Let $u = 3 + x^{5}$ . Then $d u = 5 x^{4} d x$ , so $\frac{1}{5} d u = x^{4} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

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

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Step 2.1.1

Differentiate $3 + x^{5}$ .

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

Step 2.1.2

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

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

Step 2.1.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{5}]$

Step 2.1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$0 + 5 x^{4}$

Step 2.1.5

Add $0$ and $5 x^{4}$ .

$5 x^{4}$

Step 2.2

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

$u_{lower} = 3 + 0^{5}$

Step 2.3

Simplify.

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Step 2.3.1

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

$u_{lower} = 3 + 0$

Step 2.3.2

Add $3$ and $0$ .

$u_{lower} = 3$

Step 2.4

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

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

Step 2.5

Simplify.

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Step 2.5.1

One to any power is one.

$u_{upper} = 3 + 1$

Step 2.5.2

Add $3$ and $1$ .

$u_{upper} = 4$

Step 2.6

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

$u_{lower} = 3$

$u_{upper} = 4$

Step 2.7

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

$5 \int_{3}^{4} u^{4} \frac{1}{5} d u$

Step 3

Combine $u^{4}$ and $\frac{1}{5}$ .

$5 \int_{3}^{4} \frac{u^{4}}{5} d u$

Step 4

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$5 (\frac{1}{5} \int_{3}^{4} u^{4} d u)$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{5}$ and $5$ .

$\frac{5}{5} \int_{3}^{4} u^{4} d u$

Step 5.2

Cancel the common factor of $5$ .

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Step 5.2.1

Cancel the common factor.

$\frac{5}{5} \int_{3}^{4} u^{4} d u$

Step 5.2.2

Rewrite the expression.

$1 \int_{3}^{4} u^{4} d u$

Step 5.3

Multiply $\int_{3}^{4} u^{4} d u$ by $1$ .

$\int_{3}^{4} u^{4} d u$

Step 6

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$\frac{1}{5} u^{5}]_{3}^{4}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{1}{5} u^{5}$ at $4$ and at $3$ .

$(\frac{1}{5} \cdot 4^{5}) - \frac{1}{5} \cdot 3^{5}$

Step 7.2

Simplify.

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Step 7.2.1

Raise $4$ to the power of $5$ .

$\frac{1}{5} \cdot 1024 - \frac{1}{5} \cdot 3^{5}$

Step 7.2.2

Combine $\frac{1}{5}$ and $1024$ .

$\frac{1024}{5} - \frac{1}{5} \cdot 3^{5}$

Step 7.2.3

Raise $3$ to the power of $5$ .

$\frac{1024}{5} - \frac{1}{5} \cdot 243$

Step 7.2.4

Multiply $243$ by $- 1$ .

$\frac{1024}{5} - 243 (\frac{1}{5})$

Step 7.2.5

Combine $- 243$ and $\frac{1}{5}$ .

$\frac{1024}{5} + \frac{- 243}{5}$

Step 7.2.6

Move the negative in front of the fraction.

$\frac{1024}{5} - \frac{243}{5}$

Step 7.2.7

Combine the numerators over the common denominator.

$\frac{1024 - 243}{5}$

Step 7.2.8

Subtract $243$ from $1024$ .

$\frac{781}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{781}{5}$

Decimal Form:

$156.2$

Mixed Number Form:

$156 \frac{1}{5}$

Step 9