Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $1 + \sin (θ)$ .

$\frac{d}{d θ} [1 + \sin (θ)]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

$0 + \frac{d}{d θ} [\sin (θ)]$

Step 1.1.3

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

$0 + \cos (θ)$

Step 1.1.4

Add $0$ and $\cos (θ)$ .

$\cos (θ)$

Step 1.2

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

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

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 $θ$ in $u = 1 + \sin (θ)$ .

$u_{upper} = 1 + \sin (\frac{π}{2})$

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.

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

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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

Step 2.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $2$ by $2^{\frac{1}{2}}$ by adding the exponents.

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Step 5.1

Multiply $2$ by $2^{\frac{1}{2}}$ .

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.2

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

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

Step 5.3

Combine the numerators over the common denominator.

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

Step 5.4

Add $2$ and $1$ .

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

Step 6

One to any power is one.

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

Step 7

Multiply $- 2$ by $1$ .

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.82842712 \dots$