Calculus Examples

$\int_{0}^{\frac{π}{3}} \sin (x) \cos^{3} (x) d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $\cos (x)$ .

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

Step 1.1.2

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

$- \sin (x)$

Step 1.2

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

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

Step 1.3

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

$u_{lower} = 1$

Step 1.4

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

$u_{upper} = \cos (\frac{π}{3})$

Step 1.5

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$u_{upper} = \frac{1}{2}$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = \frac{1}{2}$

Step 1.7

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

$\int_{1}^{\frac{1}{2}} - 1 u^{3} d u$

Step 2

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int_{1}^{\frac{1}{2}} u^{3} d u$

Step 3

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

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

Step 4

Simplify the expression.

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Step 4.1

Evaluate $\frac{1}{4} u^{4}$ at $\frac{1}{2}$ and at $1$ .

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

Step 4.2

Simplify.

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Step 4.2.1

Rewrite $\frac{1}{4}$ as $4^{- 1}$ .

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

Step 4.2.2

Rewrite $4$ as $2^{2}$ .

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

Step 4.2.3

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

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

Step 4.2.4

Multiply $2$ by $- 1$ .

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

Step 4.2.5

Rewrite $\frac{1}{2}$ as $2^{- 1}$ .

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

Step 4.2.6

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

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

Step 4.2.7

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

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

Step 4.2.8

Multiply $- 1$ by $1$ .

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

Step 4.2.9

Multiply the exponents in ${(2^{- 1})}^{4}$ .

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Step 4.2.9.1

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

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

Step 4.2.9.2

Multiply $- 1$ by $4$ .

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

Step 4.2.10

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

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

Step 4.2.11

Subtract $4$ from $- 2$ .

$- (2^{- 6} - \frac{1}{4} \cdot 1^{4})$

Step 4.2.12

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- (\frac{1}{2^{6}} - \frac{1}{4} \cdot 1^{4})$

Step 4.2.13

Raise $2$ to the power of $6$ .

$- (\frac{1}{64} - \frac{1}{4} \cdot 1^{4})$

Step 4.3

Simplify the expression.

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Step 4.3.1

One to any power is one.

$- (\frac{1}{64} - \frac{1}{4} \cdot 1)$

Step 4.3.2

Multiply $- 1$ by $1$ .

$- (\frac{1}{64} - \frac{1}{4})$

Step 4.4

Simplify.

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Step 4.4.1

To write $- \frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$- (\frac{1}{64} - \frac{1}{4} \cdot \frac{16}{16})$

Step 4.4.2

Write each expression with a common denominator of $64$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.2.1

Multiply $\frac{1}{4}$ by $\frac{16}{16}$ .

$- (\frac{1}{64} - \frac{16}{4 \cdot 16})$

Step 4.4.2.2

Multiply $4$ by $16$ .

$- (\frac{1}{64} - \frac{16}{64})$

Step 4.4.3

Combine the numerators over the common denominator.

$- \frac{1 - 16}{64}$

Step 4.4.4

Subtract $16$ from $1$ .

$- \frac{- 15}{64}$

Step 4.4.5

Move the negative in front of the fraction.

$- - \frac{15}{64}$

Step 4.5

Simplify.

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Step 4.5.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{15}{64})$

Step 4.5.2

Multiply $\frac{15}{64}$ by $1$ .

$\frac{15}{64}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{15}{64}$

Decimal Form:

$0.234375$