Calculus Examples

$\int_{0}^{1} {(x^{2} + 1)}^{10} (2 x) d x$

Step 1

Move $2$ to the left of ${(x^{2} + 1)}^{10}$ .

$\int_{0}^{1} 2 {(x^{2} + 1)}^{10} x d x$

Step 2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{0}^{1} {(x^{2} + 1)}^{10} x d x$

Step 3

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

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Differentiate $x^{2} + 1$ .

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

Step 3.1.2

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

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

Step 3.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 3.1.4

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

$2 x + 0$

Step 3.1.5

Add $2 x$ and $0$ .

$2 x$

Step 3.2

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

$u_{lower} = 0^{2} + 1$

Step 3.3

Simplify.

Tap for more steps...

Step 3.3.1

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

$u_{lower} = 0 + 1$

Step 3.3.2

Add $0$ and $1$ .

$u_{lower} = 1$

Step 3.4

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

$u_{upper} = 1^{2} + 1$

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

One to any power is one.

$u_{upper} = 1 + 1$

Step 3.5.2

Add $1$ and $1$ .

$u_{upper} = 2$

Step 3.6

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

$u_{lower} = 1$

$u_{upper} = 2$

Step 3.7

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

$2 \int_{1}^{2} u^{10} \frac{1}{2} d u$

Step 4

Combine $u^{10}$ and $\frac{1}{2}$ .

$2 \int_{1}^{2} \frac{u^{10}}{2} d u$

Step 5

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$2 (\frac{1}{2} \int_{1}^{2} u^{10} d u)$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Combine $\frac{1}{2}$ and $2$ .

$\frac{2}{2} \int_{1}^{2} u^{10} d u$

Step 6.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.1

Cancel the common factor.

$\frac{2}{2} \int_{1}^{2} u^{10} d u$

Step 6.2.2

Rewrite the expression.

$1 \int_{1}^{2} u^{10} d u$

Step 6.3

Multiply $\int_{1}^{2} u^{10} d u$ by $1$ .

$\int_{1}^{2} u^{10} d u$

Step 7

By the Power Rule, the integral of $u^{10}$ with respect to $u$ is $\frac{1}{11} u^{11}$ .

$\frac{1}{11} u^{11}]_{1}^{2}$

Step 8

Substitute and simplify.

Tap for more steps...

Step 8.1

Evaluate $\frac{1}{11} u^{11}$ at $2$ and at $1$ .

$(\frac{1}{11} \cdot 2^{11}) - \frac{1}{11} \cdot 1^{11}$

Step 8.2

Simplify.

Tap for more steps...

Step 8.2.1

Raise $2$ to the power of $11$ .

$\frac{1}{11} \cdot 2048 - \frac{1}{11} \cdot 1^{11}$

Step 8.2.2

Combine $\frac{1}{11}$ and $2048$ .

$\frac{2048}{11} - \frac{1}{11} \cdot 1^{11}$

Step 8.2.3

One to any power is one.

$\frac{2048}{11} - \frac{1}{11} \cdot 1$

Step 8.2.4

Multiply $- 1$ by $1$ .

$\frac{2048}{11} - \frac{1}{11}$

Step 8.2.5

Combine the numerators over the common denominator.

$\frac{2048 - 1}{11}$

Step 8.2.6

Subtract $1$ from $2048$ .

$\frac{2047}{11}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{2047}{11}$

Decimal Form:

$186.\overline{09}$

Mixed Number Form:

$186 \frac{1}{11}$

Step 10