Calculus Examples

$\int \frac{2 r}{{(1 - r)}^{7}} d r$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $1 - r$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

Rewrite the problem using $u$ and $d u$ .

$\int \frac{- 2 (- u + 1)}{u^{7}} d u$

Step 2

Move the negative in front of the fraction.

$\int - \frac{2 (- u + 1)}{u^{7}} d u$

Step 3

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

$- \int \frac{2 (- u + 1)}{u^{7}} d u$

Step 4

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$- (2 \int \frac{- u + 1}{u^{7}} d u)$

Step 5

Simplify the expression.

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Step 5.1

Multiply $2$ by $- 1$ .

$- 2 \int \frac{- u + 1}{u^{7}} d u$

Step 5.2

Move $u^{7}$ out of the denominator by raising it to the $- 1$ power.

$- 2 \int (- u + 1) {(u^{7})}^{- 1} d u$

Step 5.3

Multiply the exponents in ${(u^{7})}^{- 1}$ .

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Step 5.3.1

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

$- 2 \int (- u + 1) u^{7 \cdot - 1} d u$

Step 5.3.2

Multiply $7$ by $- 1$ .

$- 2 \int (- u + 1) u^{- 7} d u$

Step 6

Multiply $(- u + 1) u^{- 7}$ .

$- 2 \int - u \cdot u^{- 7} + 1 u^{- 7} d u$

Step 7

Simplify.

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Step 7.1

Multiply $u$ by $u^{- 7}$ by adding the exponents.

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Step 7.1.1

Move $u^{- 7}$ .

$- 2 \int - (u^{- 7} u) + 1 u^{- 7} d u$

Step 7.1.2

Multiply $u^{- 7}$ by $u$ .

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Step 7.1.2.1

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

$- 2 \int - (u^{- 7} u^{1}) + 1 u^{- 7} d u$

Step 7.1.2.2

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

$- 2 \int - u^{- 7 + 1} + 1 u^{- 7} d u$

Step 7.1.3

Add $- 7$ and $1$ .

$- 2 \int - u^{- 6} + 1 u^{- 7} d u$

Step 7.2

Multiply $u^{- 7}$ by $1$ .

$- 2 \int - u^{- 6} + u^{- 7} d u$

Step 8

Split the single integral into multiple integrals.

$- 2 (\int - u^{- 6} d u + \int u^{- 7} d u)$

Step 9

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

$- 2 (- \int u^{- 6} d u + \int u^{- 7} d u)$

Step 10

By the Power Rule, the integral of $u^{- 6}$ with respect to $u$ is $- \frac{1}{5} u^{- 5}$ .

$- 2 (- (- \frac{1}{5} u^{- 5} + C) + \int u^{- 7} d u)$

Step 11

Simplify.

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Step 11.1

Combine $u^{- 5}$ and $\frac{1}{5}$ .

$- 2 (- (- \frac{u^{- 5}}{5} + C) + \int u^{- 7} d u)$

Step 11.2

Move $u^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 2 (- (- \frac{1}{5 u^{5}} + C) + \int u^{- 7} d u)$

Step 12

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

$- 2 (- (- \frac{1}{5 u^{5}} + C) - \frac{1}{6} u^{- 6} + C)$

Step 13

Simplify.

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Step 13.1

Simplify.

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Step 13.1.1

Combine $u^{- 6}$ and $\frac{1}{6}$ .

$- 2 (- (- \frac{1}{5 u^{5}} + C) - \frac{u^{- 6}}{6} + C)$

Step 13.1.2

Move $u^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 2 (- (- \frac{1}{5 u^{5}} + C) - \frac{1}{6 u^{6}} + C)$

Step 13.2

Simplify.

$- 2 (\frac{1}{5 u^{5}} - \frac{1}{6 u^{6}}) + C$

Step 14

Replace all occurrences of $u$ with $1 - r$ .

$- 2 (\frac{1}{5 {(1 - r)}^{5}} - \frac{1}{6 {(1 - r)}^{6}}) + C$