Calculus Examples

$\int_{0}^{1} \sqrt{t^{4} + 7 t} (4 t^{3} + 7) d t$

Step 1

Let $u = t^{4} + 7 t$ . Then $d u = (4 t^{3} + 7) d t$ , so $\frac{1}{4 t^{3} + 7} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = t^{4} + 7 t$ . Find $\frac{d u}{d t}$ .

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Step 1.1.1

Differentiate $t^{4} + 7 t$ .

$\frac{d}{d t} [t^{4} + 7 t]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $t^{4} + 7 t$ with respect to $t$ is $\frac{d}{d t} [t^{4}] + \frac{d}{d t} [7 t]$ .

$\frac{d}{d t} [t^{4}] + \frac{d}{d t} [7 t]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 4$ .

$4 t^{3} + \frac{d}{d t} [7 t]$

Step 1.1.3

Evaluate $\frac{d}{d t} [7 t]$ .

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Step 1.1.3.1

Since $7$ is constant with respect to $t$ , the derivative of $7 t$ with respect to $t$ is $7 \frac{d}{d t} [t]$ .

$4 t^{3} + 7 \frac{d}{d t} [t]$

Step 1.1.3.2

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

$4 t^{3} + 7 \cdot 1$

Step 1.1.3.3

Multiply $7$ by $1$ .

$4 t^{3} + 7$

Step 1.2

Substitute the lower limit in for $t$ in $u = t^{4} + 7 t$ .

$u_{lower} = 0^{4} + 7 \cdot 0$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

$u_{lower} = 0 + 7 \cdot 0$

Step 1.3.1.2

Multiply $7$ by $0$ .

$u_{lower} = 0 + 0$

Step 1.3.2

Add $0$ and $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $t$ in $u = t^{4} + 7 t$ .

$u_{upper} = 1^{4} + 7 \cdot 1$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

One to any power is one.

$u_{upper} = 1 + 7 \cdot 1$

Step 1.5.1.2

Multiply $7$ by $1$ .

$u_{upper} = 1 + 7$

Step 1.5.2

Add $1$ and $7$ .

$u_{upper} = 8$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 8$

Step 1.7

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

$\int_{0}^{8} \sqrt{u} d u$

Step 2

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

$\int_{0}^{8} u^{\frac{1}{2}} d u$

Step 3

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

$\frac{2}{3} u^{\frac{3}{2}}]_{0}^{8}$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $8$ and at $0$ .

$(\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$\frac{2 \cdot 8^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 4.2.2

Rewrite $8$ as $2^{3}$ .

$\frac{2 \cdot {(2^{3})}^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 4.2.3

Multiply the exponents in ${(2^{3})}^{\frac{3}{2}}$ .

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Step 4.2.3.1

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

$\frac{2 \cdot 2^{3 (\frac{3}{2})}}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 4.2.3.2

Multiply $3 (\frac{3}{2})$ .

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Step 4.2.3.2.1

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

$\frac{2 \cdot 2^{\frac{3 \cdot 3}{2}}}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 4.2.3.2.2

Multiply $3$ by $3$ .

$\frac{2 \cdot 2^{\frac{9}{2}}}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 4.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2^{1 + \frac{9}{2}}}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 4.2.5

Write $1$ as a fraction with a common denominator.

$\frac{2^{\frac{2}{2} + \frac{9}{2}}}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 4.2.6

Combine the numerators over the common denominator.

$\frac{2^{\frac{2 + 9}{2}}}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 4.2.7

Add $2$ and $9$ .

$\frac{2^{\frac{11}{2}}}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 4.2.8

Rewrite $0$ as $0^{2}$ .

$\frac{2^{\frac{11}{2}}}{3} - \frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}}$

Step 4.2.9

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

$\frac{2^{\frac{11}{2}}}{3} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})}$

Step 4.2.10

Cancel the common factor of $2$ .

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Step 4.2.10.1

Cancel the common factor.

$\frac{2^{\frac{11}{2}}}{3} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})}$

Step 4.2.10.2

Rewrite the expression.

$\frac{2^{\frac{11}{2}}}{3} - \frac{2}{3} \cdot 0^{3}$

Step 4.2.11

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

$\frac{2^{\frac{11}{2}}}{3} - \frac{2}{3} \cdot 0$

Step 4.2.12

Multiply $0$ by $- 1$ .

$\frac{2^{\frac{11}{2}}}{3} + 0 (\frac{2}{3})$

Step 4.2.13

Multiply $0$ by $\frac{2}{3}$ .

$\frac{2^{\frac{11}{2}}}{3} + 0$

Step 4.2.14

Add $\frac{2^{\frac{11}{2}}}{3}$ and $0$ .

$\frac{2^{\frac{11}{2}}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2^{\frac{11}{2}}}{3}$

Decimal Form:

$15.08494466 \dots$

Step 6