Calculus Examples

$\int_{0}^{0.5} x {(1 - x)}^{3} d x$

Step 1

Let $u = 1 - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 1 - x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $1 - x$ .

$\frac{d}{d x} [1 - x]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $1 - x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- x]$

Step 1.1.2.2

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

$0 + \frac{d}{d x} [- x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- x]$ .

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$0 - \frac{d}{d x} [x]$

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 $n = 1$ .

$0 - 1 \cdot 1$

Step 1.1.3.3

Multiply $- 1$ by $1$ .

$0 - 1$

Step 1.1.4

Subtract $1$ from $0$ .

$- 1$

Step 1.2

Substitute the lower limit in for $x$ in $u = 1 - x$ .

$u_{lower} = 1 - 0$

Step 1.3

Subtract $0$ from $1$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 - x$ .

$u_{upper} = 1 - 1 \cdot 0.5$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 1$ by $0.5$ .

$u_{upper} = 1 - 0.5$

Step 1.5.2

Subtract $0.5$ from $1$ .

$u_{upper} = 0.5$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 0.5$

Step 1.7

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

$\int_{1}^{0.5} - (- u + 1) u^{3} d u$

Step 2

Multiply $- (- u + 1) u^{3}$ .

$\int_{1}^{0.5} - - u u^{3} - 1 \cdot 1 u^{3} d u$

Step 3

Simplify.

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Step 3.1

Multiply $- 1$ by $- 1$ .

$\int_{1}^{0.5} 1 u \cdot u^{3} - 1 \cdot 1 u^{3} d u$

Step 3.2

Multiply $u$ by $1$ .

$\int_{1}^{0.5} u \cdot u^{3} - 1 \cdot 1 u^{3} d u$

Step 3.3

Multiply $u$ by $u^{3}$ by adding the exponents.

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Step 3.3.1

Multiply $u$ by $u^{3}$ .

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Step 3.3.1.1

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

$\int_{1}^{0.5} u^{1} u^{3} - 1 \cdot 1 u^{3} d u$

Step 3.3.1.2

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

$\int_{1}^{0.5} u^{1 + 3} - 1 \cdot 1 u^{3} d u$

Step 3.3.2

Add $1$ and $3$ .

$\int_{1}^{0.5} u^{4} - 1 \cdot 1 u^{3} d u$

Step 3.4

Multiply $- 1$ by $1$ .

$\int_{1}^{0.5} u^{4} - 1 u^{3} d u$

Step 3.5

Rewrite $- 1 u^{3}$ as $- u^{3}$ .

$\int_{1}^{0.5} u^{4} - u^{3} d u$

Step 4

Split the single integral into multiple integrals.

$\int_{1}^{0.5} u^{4} d u + \int_{1}^{0.5} - u^{3} d u$

Step 5

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$\frac{1}{5} u^{5}]_{1}^{0.5} + \int_{1}^{0.5} - u^{3} d u$

Step 6

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

$\frac{1}{5} u^{5}]_{1}^{0.5} - \int_{1}^{0.5} u^{3} d u$

Step 7

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

$\frac{1}{5} u^{5}]_{1}^{0.5} - (\frac{1}{4} u^{4}]_{1}^{0.5})$

Step 8

Combine $\frac{1}{4}$ and $u^{4}$ .

$\frac{1}{5} u^{5}]_{1}^{0.5} - (\frac{u^{4}}{4}]_{1}^{0.5})$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{1}{5} u^{5}$ at $0.5$ and at $1$ .

$(\frac{1}{5} \cdot {0.5}^{5}) - \frac{1}{5} \cdot 1^{5} - (\frac{u^{4}}{4}]_{1}^{0.5})$

Step 9.2

Evaluate $\frac{u^{4}}{4}$ at $0.5$ and at $1$ .

$\frac{1}{5} \cdot {0.5}^{5} - \frac{1}{5} \cdot 1^{5} - (\frac{{0.5}^{4}}{4} - \frac{1^{4}}{4})$

Step 9.3

Simplify.

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Step 9.3.1

Raise $0.5$ to the power of $5$ .

$\frac{1}{5} \cdot 0.03125 - \frac{1}{5} \cdot 1^{5} - (\frac{{0.5}^{4}}{4} - \frac{1^{4}}{4})$

Step 9.3.2

Combine $\frac{1}{5}$ and $0.03125$ .

$\frac{0.03125}{5} - \frac{1}{5} \cdot 1^{5} - (\frac{{0.5}^{4}}{4} - \frac{1^{4}}{4})$

Step 9.3.3

One to any power is one.

$\frac{0.03125}{5} - \frac{1}{5} \cdot 1 - (\frac{{0.5}^{4}}{4} - \frac{1^{4}}{4})$

Step 9.3.4

Multiply $- 1$ by $1$ .

$\frac{0.03125}{5} - \frac{1}{5} - (\frac{{0.5}^{4}}{4} - \frac{1^{4}}{4})$

Step 9.3.5

Combine the numerators over the common denominator.

$\frac{0.03125 - 1}{5} - (\frac{{0.5}^{4}}{4} - \frac{1^{4}}{4})$

Step 9.3.6

Subtract $1$ from $0.03125$ .

$\frac{- 0.96875}{5} - (\frac{{0.5}^{4}}{4} - \frac{1^{4}}{4})$

Step 9.3.7

Move the negative in front of the fraction.

$- \frac{0.96875}{5} - (\frac{{0.5}^{4}}{4} - \frac{1^{4}}{4})$

Step 9.3.8

Raise $0.5$ to the power of $4$ .

$- \frac{0.96875}{5} - (\frac{0.0625}{4} - \frac{1^{4}}{4})$

Step 9.3.9

One to any power is one.

$- \frac{0.96875}{5} - (\frac{0.0625}{4} - \frac{1}{4})$

Step 9.3.10

Combine the numerators over the common denominator.

$- \frac{0.96875}{5} - \frac{0.0625 - 1}{4}$

Step 9.3.11

Subtract $1$ from $0.0625$ .

$- \frac{0.96875}{5} - \frac{- 0.9375}{4}$

Step 9.3.12

Move the negative in front of the fraction.

$- \frac{0.96875}{5} - - \frac{0.9375}{4}$

Step 9.3.13

Multiply $- 1$ by $- 1$ .

$- \frac{0.96875}{5} + 1 (\frac{0.9375}{4})$

Step 9.3.14

Multiply $\frac{0.9375}{4}$ by $1$ .

$- \frac{0.96875}{5} + \frac{0.9375}{4}$

Step 9.3.15

To write $- \frac{0.96875}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{0.96875}{5} \cdot \frac{4}{4} + \frac{0.9375}{4}$

Step 9.3.16

To write $\frac{0.9375}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{0.96875}{5} \cdot \frac{4}{4} + \frac{0.9375}{4} \cdot \frac{5}{5}$

Step 9.3.17

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.3.17.1

Multiply $\frac{0.96875}{5}$ by $\frac{4}{4}$ .

$- \frac{0.96875 \cdot 4}{5 \cdot 4} + \frac{0.9375}{4} \cdot \frac{5}{5}$

Step 9.3.17.2

Multiply $5$ by $4$ .

$- \frac{0.96875 \cdot 4}{20} + \frac{0.9375}{4} \cdot \frac{5}{5}$

Step 9.3.17.3

Multiply $\frac{0.9375}{4}$ by $\frac{5}{5}$ .

$- \frac{0.96875 \cdot 4}{20} + \frac{0.9375 \cdot 5}{4 \cdot 5}$

Step 9.3.17.4

Multiply $4$ by $5$ .

$- \frac{0.96875 \cdot 4}{20} + \frac{0.9375 \cdot 5}{20}$

Step 9.3.18

Combine the numerators over the common denominator.

$\frac{- 0.96875 \cdot 4 + 0.9375 \cdot 5}{20}$

Step 9.3.19

Multiply $- 0.96875$ by $4$ .

$\frac{- 3.875 + 0.9375 \cdot 5}{20}$

Step 9.3.20

Multiply $0.9375$ by $5$ .

$\frac{- 3.875 + 4.6875}{20}$

Step 9.3.21

Add $- 3.875$ and $4.6875$ .

$\frac{0.8125}{20}$

Step 10

Divide $0.8125$ by $20$ .

$0.040625$

Step 11