Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $x^{2} + x$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

$2 x + 1$

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

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

$u_{lower} = 0 + 0$

Step 1.3.2

Add $0$ and $0$ .

$u_{lower} = 0$

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

One to any power is one.

$u_{upper} = 1 + 1$

Step 1.5.2

Add $1$ and $1$ .

$u_{upper} = 2$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 2$

Step 1.7

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

$\int_{0}^{2} \sqrt{u} d u$

Step 2

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

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

Step 3

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

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

Step 4

Substitute and simplify.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

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

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

Step 4.2.2

Multiply $2$ by $2^{\frac{3}{2}}$ by adding the exponents.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

Raise $2$ to the power of $1$ .

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

Step 4.2.2.1.2

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

Combine the numerators over the common denominator.

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

Step 4.2.2.4

Add $2$ and $3$ .

$\frac{2^{\frac{5}{2}}}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 4.2.3

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

$\frac{2^{\frac{5}{2}}}{3} - \frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}}$

Step 4.2.4

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

$\frac{2^{\frac{5}{2}}}{3} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})}$

Step 4.2.5

Cancel the common factor of $2$ .

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Step 4.2.5.1

Cancel the common factor.

$\frac{2^{\frac{5}{2}}}{3} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})}$

Step 4.2.5.2

Rewrite the expression.

$\frac{2^{\frac{5}{2}}}{3} - \frac{2}{3} \cdot 0^{3}$

Step 4.2.6

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

$\frac{2^{\frac{5}{2}}}{3} - \frac{2}{3} \cdot 0$

Step 4.2.7

Multiply $0$ by $- 1$ .

$\frac{2^{\frac{5}{2}}}{3} + 0 (\frac{2}{3})$

Step 4.2.8

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

$\frac{2^{\frac{5}{2}}}{3} + 0$

Step 4.2.9

Add $\frac{2^{\frac{5}{2}}}{3}$ and $0$ .

$\frac{2^{\frac{5}{2}}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2^{\frac{5}{2}}}{3}$

Decimal Form:

$1.88561808 \dots$

Step 6