Calculus Examples

$\int_{0}^{\frac{π}{4}} 4 \sin (x) \cos (x) d x$

Step 1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{0}^{\frac{π}{4}} \sin (x) \cos (x) d x$

Step 2

Let $u = \sin (x)$ . 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 2.1

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

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Step 2.1.1

Differentiate $\sin (x)$ .

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

Step 2.1.2

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

$\cos (x)$

Step 2.2

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

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

Step 2.3

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

$u_{lower} = 0$

Step 2.4

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

$u_{upper} = \sin (\frac{π}{4})$

Step 2.5

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$u_{upper} = \frac{\sqrt{2}}{2}$

Step 2.6

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

$u_{lower} = 0$

$u_{upper} = \frac{\sqrt{2}}{2}$

Step 2.7

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

$4 \int_{0}^{\frac{\sqrt{2}}{2}} u d u$

Step 3

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

$4 (\frac{1}{2} u^{2}]_{0}^{\frac{\sqrt{2}}{2}})$

Step 4

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

$4 (\frac{u^{2}}{2}]_{0}^{\frac{\sqrt{2}}{2}})$

Step 5

Substitute and simplify.

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Step 5.1

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

$4 ((\frac{{(\frac{\sqrt{2}}{2})}^{2}}{2}) - \frac{0^{2}}{2})$

Step 5.2

Simplify.

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Step 5.2.1

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

$4 (\frac{{(\frac{\sqrt{2}}{2})}^{2}}{2} - \frac{0}{2})$

Step 5.2.2

Cancel the common factor of $0$ and $2$ .

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Step 5.2.2.1

Factor $2$ out of $0$ .

$4 (\frac{{(\frac{\sqrt{2}}{2})}^{2}}{2} - \frac{2 (0)}{2})$

Step 5.2.2.2

Cancel the common factors.

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Step 5.2.2.2.1

Factor $2$ out of $2$ .

$4 (\frac{{(\frac{\sqrt{2}}{2})}^{2}}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 5.2.2.2.2

Cancel the common factor.

$4 (\frac{{(\frac{\sqrt{2}}{2})}^{2}}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 5.2.2.2.3

Rewrite the expression.

$4 (\frac{{(\frac{\sqrt{2}}{2})}^{2}}{2} - \frac{0}{1})$

Step 5.2.2.2.4

Divide $0$ by $1$ .

$4 (\frac{{(\frac{\sqrt{2}}{2})}^{2}}{2} - 0)$

Step 5.2.3

Multiply $- 1$ by $0$ .

$4 (\frac{{(\frac{\sqrt{2}}{2})}^{2}}{2} + 0)$

Step 5.2.4

Add $\frac{{(\frac{\sqrt{2}}{2})}^{2}}{2}$ and $0$ .

$4 \frac{{(\frac{\sqrt{2}}{2})}^{2}}{2}$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $2$ .

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Step 6.1.1

Factor $2$ out of $4$ .

$2 (2) \frac{{(\frac{\sqrt{2}}{2})}^{2}}{2}$

Step 6.1.2

Cancel the common factor.

$2 \cdot 2 \frac{{(\frac{\sqrt{2}}{2})}^{2}}{2}$

Step 6.1.3

Rewrite the expression.

$2 {(\frac{\sqrt{2}}{2})}^{2}$

Step 6.2

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$2 \frac{{\sqrt{2}}^{2}}{2^{2}}$

Step 6.3

Cancel the common factor of $2$ .

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Step 6.3.1

Factor $2$ out of $2^{2}$ .

$2 \frac{{\sqrt{2}}^{2}}{2 \cdot 2}$

Step 6.3.2

Cancel the common factor.

$2 \frac{{\sqrt{2}}^{2}}{2 \cdot 2}$

Step 6.3.3

Rewrite the expression.

$\frac{{\sqrt{2}}^{2}}{2}$

Step 6.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 6.4.1

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

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

Step 6.4.2

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

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

Step 6.4.3

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

$\frac{2^{\frac{2}{2}}}{2}$

Step 6.4.4

Cancel the common factor of $2$ .

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Step 6.4.4.1

Cancel the common factor.

$\frac{2^{\frac{2}{2}}}{2}$

Step 6.4.4.2

Rewrite the expression.

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

Step 6.4.5

Evaluate the exponent.

$\frac{2}{2}$

Step 6.5

Divide $2$ by $2$ .

$1$