Calculus Examples

$\int_{0}^{π} (\sin^{6} (4 x)) \cos (4 x) d x$

Step 1

Let $u_{2} = \sin (4 x)$ . Then $d u_{2} = 4 \cos (4 x) d x$ , so $\frac{1}{4} d u_{2} = \cos (4 x) d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = \sin (4 x)$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $\sin (4 x)$ .

$\frac{d}{d x} [\sin (4 x)]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 4 x$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $4 x$ .

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [4 x]$

Step 1.1.2.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cos (u_{1}) \frac{d}{d x} [4 x]$

Step 1.1.2.3

Replace all occurrences of $u_{1}$ with $4 x$ .

$\cos (4 x) \frac{d}{d x} [4 x]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\cos (4 x) (4 \frac{d}{d x} [x])$

Step 1.1.3.2

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

$\cos (4 x) (4 \cdot 1)$

Step 1.1.3.3

Simplify the expression.

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Step 1.1.3.3.1

Multiply $4$ by $1$ .

$\cos (4 x) \cdot 4$

Step 1.1.3.3.2

Move $4$ to the left of $\cos (4 x)$ .

$4 \cos (4 x)$

Step 1.2

Substitute the lower limit in for $x$ in $u_{2} = \sin (4 x)$ .

$u_{lower} = \sin (4 \cdot 0)$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $4$ by $0$ .

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

Step 1.3.2

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

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u_{2} = \sin (4 x)$ .

$u_{upper} = \sin (4 π)$

Step 1.5

Simplify.

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Step 1.5.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

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

Step 1.5.2

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

$u_{upper} = 0$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 0$

Step 1.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\int_{0}^{0} {u_{2}}^{6} \frac{1}{4} d u_{2}$

Step 2

Combine ${u_{2}}^{6}$ and $\frac{1}{4}$ .

$\int_{0}^{0} \frac{{u_{2}}^{6}}{4} d u_{2}$

Step 3

Since $\frac{1}{4}$ is constant with respect to $u_{2}$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int_{0}^{0} {u_{2}}^{6} d u_{2}$

Step 4

By the Power Rule, the integral of ${u_{2}}^{6}$ with respect to $u_{2}$ is $\frac{1}{7} {u_{2}}^{7}$ .

$\frac{1}{4} \frac{1}{7} {u_{2}}^{7}]_{0}^{0}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{1}{7} {u_{2}}^{7}$ at $0$ and at $0$ .

$\frac{1}{4} ((\frac{1}{7} \cdot 0^{7}) - \frac{1}{7} \cdot 0^{7})$

Step 5.2

Simplify.

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Step 5.2.1

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

$\frac{1}{4} (\frac{1}{7} \cdot 0 - \frac{1}{7} \cdot 0^{7})$

Step 5.2.2

Multiply $\frac{1}{7}$ by $0$ .

$\frac{1}{4} (0 - \frac{1}{7} \cdot 0^{7})$

Step 5.2.3

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

$\frac{1}{4} (0 - \frac{1}{7} \cdot 0)$

Step 5.2.4

Multiply $0$ by $- 1$ .

$\frac{1}{4} (0 + 0 (\frac{1}{7}))$

Step 5.2.5

Multiply $0$ by $\frac{1}{7}$ .

$\frac{1}{4} (0 + 0)$

Step 5.2.6

Add $0$ and $0$ .

$\frac{1}{4} \cdot 0$

Step 5.2.7

Multiply $\frac{1}{4}$ by $0$ .

$0$