Calculus Examples

$\int_{0}^{\frac{π}{2}} \cos (x) \sqrt{1 + \sin (x)} d x$

Step 1

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

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Step 1.1

Let $u = 1 + \sin (x)$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $1 + \sin (x)$ .

$\frac{d}{d x} [1 + \sin (x)]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

$\frac{d}{d x} [1] + \frac{d}{d x} [\sin (x)]$

Step 1.1.2.2

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

$0 + \frac{d}{d x} [\sin (x)]$

Step 1.1.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$0 + \cos (x)$

Step 1.1.4

Add $0$ and $\cos (x)$ .

$\cos (x)$

Step 1.2

Substitute the lower limit in for $x$ in $u = 1 + \sin (x)$ .

$u_{lower} = 1 + \sin (0)$

Step 1.3

Simplify.

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Step 1.3.1

The exact value of $\sin (0)$ is $0$ .

$u_{lower} = 1 + 0$

Step 1.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 + \sin (x)$ .

$u_{upper} = 1 + \sin (\frac{π}{2})$

Step 1.5

Simplify.

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Step 1.5.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$u_{upper} = 1 + 1$

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.

$u_{lower} = 1$

$u_{upper} = 2$

Step 1.7

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

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

Step 2

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

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

Step 3

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

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

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $2$ and at $1$ .

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

Step 4.2

Simplify.

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Step 4.2.1

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

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

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 $1$ .

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

Step 4.2.2.1.2

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

Combine the numerators over the common denominator.

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

Step 4.2.2.4

Add $2$ and $3$ .

$\frac{2^{\frac{5}{2}}}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 4.2.3

One to any power is one.

$\frac{2^{\frac{5}{2}}}{3} - \frac{2}{3} \cdot 1$

Step 4.2.4

Multiply $- 1$ by $1$ .

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

Step 4.2.5

Combine the numerators over the common denominator.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.21895141 \dots$