Calculus Examples

$\int_{0}^{1} {(3 t - 1)}^{50} d t$

Step 1

Let $u = 3 t - 1$ . Then $d u = 3 d t$ , so $\frac{1}{3} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 3 t - 1$ . Find $\frac{d u}{d t}$ .

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Step 1.1.1

Differentiate $3 t - 1$ .

$\frac{d}{d t} [3 t - 1]$

Step 1.1.2

By the Sum Rule, the derivative of $3 t - 1$ with respect to $t$ is $\frac{d}{d t} [3 t] + \frac{d}{d t} [- 1]$ .

$\frac{d}{d t} [3 t] + \frac{d}{d t} [- 1]$

Step 1.1.3

Evaluate $\frac{d}{d t} [3 t]$ .

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Step 1.1.3.1

Since $3$ is constant with respect to $t$ , the derivative of $3 t$ with respect to $t$ is $3 \frac{d}{d t} [t]$ .

$3 \frac{d}{d t} [t] + \frac{d}{d t} [- 1]$

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 $n = 1$ .

$3 \cdot 1 + \frac{d}{d t} [- 1]$

Step 1.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d t} [- 1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

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

$3 + 0$

Step 1.1.4.2

Add $3$ and $0$ .

$3$

Step 1.2

Substitute the lower limit in for $t$ in $u = 3 t - 1$ .

$u_{lower} = 3 \cdot 0 - 1$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $3$ by $0$ .

$u_{lower} = 0 - 1$

Step 1.3.2

Subtract $1$ from $0$ .

$u_{lower} = - 1$

Step 1.4

Substitute the upper limit in for $t$ in $u = 3 t - 1$ .

$u_{upper} = 3 \cdot 1 - 1$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $3$ by $1$ .

$u_{upper} = 3 - 1$

Step 1.5.2

Subtract $1$ from $3$ .

$u_{upper} = 2$

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.

$\int_{- 1}^{2} u^{50} \frac{1}{3} d u$

Step 2

Combine $u^{50}$ and $\frac{1}{3}$ .

$\int_{- 1}^{2} \frac{u^{50}}{3} d u$

Step 3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int_{- 1}^{2} u^{50} d u$

Step 4

By the Power Rule, the integral of $u^{50}$ with respect to $u$ is $\frac{1}{51} u^{51}$ .

$\frac{1}{3} \frac{1}{51} u^{51}]_{- 1}^{2}$

Step 5

Combine fractions.

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Step 5.1

Evaluate $\frac{1}{51} u^{51}$ at $2$ and at $- 1$ .

$\frac{1}{3} ((\frac{1}{51} \cdot 2^{51}) - \frac{1}{51} {(- 1)}^{51})$

Step 5.2

Raise $2$ to the power of $51$ .

$\frac{1}{3} (\frac{1}{51} \cdot 2251799813685250 - \frac{1}{51} {(- 1)}^{51})$

Step 5.3

Combine $\frac{1}{51}$ and $2251799813685250$ .

$\frac{1}{3} (\frac{2251799813685250}{51} - \frac{1}{51} {(- 1)}^{51})$

Step 5.4

Simplify the expression.

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Step 5.4.1

Raise $- 1$ to the power of $51$ .

$\frac{1}{3} (\frac{2251799813685250}{51} - \frac{1}{51} \cdot - 1)$

Step 5.4.2

Simplify.

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Step 5.4.2.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{3} (\frac{2251799813685250}{51} + 1 (\frac{1}{51}))$

Step 5.4.2.2

Multiply $\frac{1}{51}$ by $1$ .

$\frac{1}{3} (\frac{2251799813685250}{51} + \frac{1}{51})$

Step 5.4.3

Simplify.

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Step 5.4.3.1

Combine the numerators over the common denominator.

$\frac{1}{3} \cdot \frac{2251799813685250 + 1}{51}$

Step 5.4.3.2

Add $2251799813685250$ and $1$ .

$\frac{1}{3} \cdot \frac{2251799813685251}{51}$

Step 5.4.4

Simplify.

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Step 5.4.4.1

Multiply $\frac{1}{3}$ by $\frac{2251799813685251}{51}$ .

$\frac{2251799813685251}{3 \cdot 51}$

Step 5.4.4.2

Multiply $3$ by $51$ .

$\frac{2251799813685251}{153}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{2251799813685251}{153}$

Decimal Form:

$14717645841080.1$