Calculus Examples

$\int_{0}^{\frac{π}{8}} 3^{\cos (4 t)} \sin (4 t) d t$

Step 1

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

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Step 1.1

Let $u_{2} = \cos (4 t)$ . Find $\frac{d u_{2}}{d t}$ .

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Step 1.1.1

Differentiate $\cos (4 t)$ .

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

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \cos (t)$ and $g (t) = 4 t$ .

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

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

Step 1.1.3.2

Multiply $4$ by $- 1$ .

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

Step 1.1.3.3

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

$- 4 \sin (4 t) \cdot 1$

Step 1.1.3.4

Multiply $- 4$ by $1$ .

$- 4 \sin (4 t)$

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

Multiply $4$ by $0$ .

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

Step 1.3.2

The exact value of $\cos (0)$ is $1$ .

$u_{lower} = 1$

Step 1.4

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

$u_{upper} = \cos (4 \frac{π}{8})$

Step 1.5

Simplify.

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Step 1.5.1

Cancel the common factor of $4$ .

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Step 1.5.1.1

Factor $4$ out of $8$ .

$u_{upper} = \cos (4 \frac{π}{4 (2)})$

Step 1.5.1.2

Cancel the common factor.

$u_{upper} = \cos (4 \frac{π}{4 \cdot 2})$

Step 1.5.1.3

Rewrite the expression.

$u_{upper} = \cos (\frac{π}{2})$

Step 1.5.2

The exact value of $\cos (\frac{π}{2})$ 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.

$u_{lower} = 1$

$u_{upper} = 0$

Step 1.7

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

$\int_{1}^{0} 3^{u_{2}} \frac{1}{- 4} d u_{2}$

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

$\int_{1}^{0} 3^{u_{2}} (- \frac{1}{4}) d u_{2}$

Step 2.2

Combine $3^{u_{2}}$ and $\frac{1}{4}$ .

$\int_{1}^{0} - \frac{3^{u_{2}}}{4} d u_{2}$

Step 3

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

$- \int_{1}^{0} \frac{3^{u_{2}}}{4} d u_{2}$

Step 4

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

$- (\frac{1}{4} \int_{1}^{0} 3^{u_{2}} d u_{2})$

Step 5

The integral of $3^{u_{2}}$ with respect to $u_{2}$ is $\frac{3^{u_{2}}}{\ln (3)}$ .

$- \frac{1}{4} \frac{3^{u_{2}}}{\ln (3)}]_{1}^{0}$

Step 6

Combine $\frac{3^{u_{2}}}{\ln (3)}]_{1}^{0}$ and $\frac{1}{4}$ .

$- \frac{\frac{3^{u_{2}}}{\ln (3)}]_{1}^{0}}{4}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{3^{u_{2}}}{\ln (3)}$ at $0$ and at $1$ .

$- \frac{(\frac{3^{0}}{\ln (3)}) - \frac{3^{1}}{\ln (3)}}{4}$

Step 7.2

Simplify.

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Step 7.2.1

Anything raised to $0$ is $1$ .

$- \frac{\frac{1}{\ln (3)} - \frac{3^{1}}{\ln (3)}}{4}$

Step 7.2.2

Evaluate the exponent.

$- \frac{\frac{1}{\ln (3)} - \frac{3}{\ln (3)}}{4}$

Step 7.2.3

Combine the numerators over the common denominator.

$- \frac{\frac{1 - 3}{\ln (3)}}{4}$

Step 7.2.4

Subtract $3$ from $1$ .

$- \frac{\frac{- 2}{\ln (3)}}{4}$

Step 7.2.5

Move the negative in front of the fraction.

$- \frac{- \frac{2}{\ln (3)}}{4}$

Step 7.2.6

Rewrite $\frac{- \frac{2}{\ln (3)}}{4}$ as a product.

$- (- \frac{2}{\ln (3)} \cdot \frac{1}{4})$

Step 7.2.7

Multiply $\frac{1}{4}$ by $\frac{2}{\ln (3)}$ .

$- - \frac{2}{4 \ln (3)}$

Step 7.2.8

Cancel the common factor of $2$ and $4$ .

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Step 7.2.8.1

Factor $2$ out of $2$ .

$- - \frac{2 (1)}{4 \ln (3)}$

Step 7.2.8.2

Cancel the common factors.

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Step 7.2.8.2.1

Factor $2$ out of $4 \ln (3)$ .

$- - \frac{2 (1)}{2 (2 \ln (3))}$

Step 7.2.8.2.2

Cancel the common factor.

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

Step 7.2.8.2.3

Rewrite the expression.

$- - \frac{1}{2 \ln (3)}$

Step 7.2.9

Multiply $- 1$ by $- 1$ .

$1 \frac{1}{2 \ln (3)}$

Step 7.2.10

Multiply $\frac{1}{2 \ln (3)}$ by $1$ .

$\frac{1}{2 \ln (3)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2 \ln (3)}$

Decimal Form:

$0.45511961 \dots$