Calculus Examples

$\int_{0}^{1} x^{2} {(1 + 2 x^{3})}^{5} d x$

Step 1

Let $u = 1 + 2 x^{3}$ . Then $d u = 6 x^{2} d x$ , so $\frac{1}{6} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = 1 + 2 x^{3}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $1 + 2 x^{3}$ .

$\frac{d}{d x} [1 + 2 x^{3}]$

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

By the Sum Rule, the derivative of $1 + 2 x^{3}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [2 x^{3}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [2 x^{3}]$

Step 1.1.2.2

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

$0 + \frac{d}{d x} [2 x^{3}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [2 x^{3}]$ .

Tap for more steps...

Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$0 + 2 \frac{d}{d x} [x^{3}]$

Step 1.1.3.2

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

$0 + 2 (3 x^{2})$

Step 1.1.3.3

Multiply $3$ by $2$ .

$0 + 6 x^{2}$

Step 1.1.4

Add $0$ and $6 x^{2}$ .

$6 x^{2}$

Step 1.2

Substitute the lower limit in for $x$ in $u = 1 + 2 x^{3}$ .

$u_{lower} = 1 + 2 \cdot 0^{3}$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

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

$u_{lower} = 1 + 2 \cdot 0$

Step 1.3.1.2

Multiply $2$ by $0$ .

$u_{lower} = 1 + 0$

Step 1.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 + 2 x^{3}$ .

$u_{upper} = 1 + 2 \cdot 1^{3}$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Simplify each term.

Tap for more steps...

Step 1.5.1.1

One to any power is one.

$u_{upper} = 1 + 2 \cdot 1$

Step 1.5.1.2

Multiply $2$ by $1$ .

$u_{upper} = 1 + 2$

Step 1.5.2

Add $1$ and $2$ .

$u_{upper} = 3$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 3$

Step 1.7

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

$\int_{1}^{3} u^{5} \frac{1}{6} d u$

Step 2

Combine $u^{5}$ and $\frac{1}{6}$ .

$\int_{1}^{3} \frac{u^{5}}{6} d u$

Step 3

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$\frac{1}{6} \int_{1}^{3} u^{5} d u$

Step 4

By the Power Rule, the integral of $u^{5}$ with respect to $u$ is $\frac{1}{6} u^{6}$ .

$\frac{1}{6} \frac{1}{6} u^{6}]_{1}^{3}$

Step 5

Simplify the expression.

Tap for more steps...

Step 5.1

Evaluate $\frac{1}{6} u^{6}$ at $3$ and at $1$ .

$\frac{1}{6} ((\frac{1}{6} \cdot 3^{6}) - \frac{1}{6} \cdot 1^{6})$

Step 5.2

Raise $3$ to the power of $6$ .

$\frac{1}{6} (\frac{1}{6} \cdot 729 - \frac{1}{6} \cdot 1^{6})$

Step 5.3

Simplify.

Tap for more steps...

Step 5.3.1

Combine $\frac{1}{6}$ and $729$ .

$\frac{1}{6} (\frac{729}{6} - \frac{1}{6} \cdot 1^{6})$

Step 5.3.2

Cancel the common factor of $729$ and $6$ .

Tap for more steps...

Step 5.3.2.1

Factor $3$ out of $729$ .

$\frac{1}{6} (\frac{3 (243)}{6} - \frac{1}{6} \cdot 1^{6})$

Step 5.3.2.2

Cancel the common factors.

Tap for more steps...

Step 5.3.2.2.1

Factor $3$ out of $6$ .

$\frac{1}{6} (\frac{3 \cdot 243}{3 \cdot 2} - \frac{1}{6} \cdot 1^{6})$

Step 5.3.2.2.2

Cancel the common factor.

$\frac{1}{6} (\frac{3 \cdot 243}{3 \cdot 2} - \frac{1}{6} \cdot 1^{6})$

Step 5.3.2.2.3

Rewrite the expression.

$\frac{1}{6} (\frac{243}{2} - \frac{1}{6} \cdot 1^{6})$

Step 5.4

Simplify the expression.

Tap for more steps...

Step 5.4.1

One to any power is one.

$\frac{1}{6} (\frac{243}{2} - \frac{1}{6} \cdot 1)$

Step 5.4.2

Multiply $- 1$ by $1$ .

$\frac{1}{6} (\frac{243}{2} - \frac{1}{6})$

Step 5.5

Simplify.

Tap for more steps...

Step 5.5.1

To write $\frac{243}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{6} (\frac{243}{2} \cdot \frac{3}{3} - \frac{1}{6})$

Step 5.5.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 5.5.2.1

Multiply $\frac{243}{2}$ by $\frac{3}{3}$ .

$\frac{1}{6} (\frac{243 \cdot 3}{2 \cdot 3} - \frac{1}{6})$

Step 5.5.2.2

Multiply $2$ by $3$ .

$\frac{1}{6} (\frac{243 \cdot 3}{6} - \frac{1}{6})$

Step 5.5.3

Combine the numerators over the common denominator.

$\frac{1}{6} \cdot \frac{243 \cdot 3 - 1}{6}$

Step 5.5.4

Simplify the numerator.

Tap for more steps...

Step 5.5.4.1

Multiply $243$ by $3$ .

$\frac{1}{6} \cdot \frac{729 - 1}{6}$

Step 5.5.4.2

Subtract $1$ from $729$ .

$\frac{1}{6} \cdot \frac{728}{6}$

Step 5.5.5

Cancel the common factor of $728$ and $6$ .

Tap for more steps...

Step 5.5.5.1

Factor $2$ out of $728$ .

$\frac{1}{6} \cdot \frac{2 (364)}{6}$

Step 5.5.5.2

Cancel the common factors.

Tap for more steps...

Step 5.5.5.2.1

Factor $2$ out of $6$ .

$\frac{1}{6} \cdot \frac{2 \cdot 364}{2 \cdot 3}$

Step 5.5.5.2.2

Cancel the common factor.

$\frac{1}{6} \cdot \frac{2 \cdot 364}{2 \cdot 3}$

Step 5.5.5.2.3

Rewrite the expression.

$\frac{1}{6} \cdot \frac{364}{3}$

Step 5.6

Simplify.

Tap for more steps...

Step 5.6.1

Multiply $\frac{1}{6}$ by $\frac{364}{3}$ .

$\frac{364}{6 \cdot 3}$

Step 5.6.2

Multiply $6$ by $3$ .

$\frac{364}{18}$

Step 5.6.3

Cancel the common factor of $364$ and $18$ .

Tap for more steps...

Step 5.6.3.1

Factor $2$ out of $364$ .

$\frac{2 (182)}{18}$

Step 5.6.3.2

Cancel the common factors.

Tap for more steps...

Step 5.6.3.2.1

Factor $2$ out of $18$ .

$\frac{2 \cdot 182}{2 \cdot 9}$

Step 5.6.3.2.2

Cancel the common factor.

$\frac{2 \cdot 182}{2 \cdot 9}$

Step 5.6.3.2.3

Rewrite the expression.

$\frac{182}{9}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{182}{9}$

Decimal Form:

$20.\overline{2}$

Mixed Number Form:

$20 \frac{2}{9}$