Calculus Examples

$\int \frac{1}{{(1 - 6 t)}^{4}} d t d t$

Step 1

Let $u = 1 - 6 t$ . Then $d u = - 6 d t$ , so $- \frac{1}{6} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 1 - 6 t$ . Find $\frac{d u}{d t}$ .

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Step 1.1.1

Differentiate $1 - 6 t$ .

$\frac{d}{d t} [1 - 6 t]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $1 - 6 t$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [- 6 t]$ .

$\frac{d}{d t} [1] + \frac{d}{d t} [- 6 t]$

Step 1.1.2.2

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

$0 + \frac{d}{d t} [- 6 t]$

Step 1.1.3

Evaluate $\frac{d}{d t} [- 6 t]$ .

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Step 1.1.3.1

Since $- 6$ is constant with respect to $t$ , the derivative of $- 6 t$ with respect to $t$ is $- 6 \frac{d}{d t} [t]$ .

$0 - 6 \frac{d}{d t} [t]$

Step 1.1.3.2

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

$0 - 6 \cdot 1$

Step 1.1.3.3

Multiply $- 6$ by $1$ .

$0 - 6$

Step 1.1.4

Subtract $6$ from $0$ .

$- 6$

Step 1.2

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

$\int \frac{1}{u^{4}} \cdot \frac{1}{- 6} d u d t$

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

$\int \frac{1}{u^{4}} (- \frac{1}{6}) d u d t$

Step 2.2

Multiply $\frac{1}{u^{4}}$ by $\frac{1}{6}$ .

$\int - \frac{1}{u^{4} \cdot 6} d u d t$

Step 2.3

Move $6$ to the left of $u^{4}$ .

$\int - \frac{1}{6 u^{4}} d u d t$

Step 3

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

$- \int \frac{1}{6 u^{4}} d u d t$

Step 4

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$- (\frac{1}{6} \int \frac{1}{u^{4}} d u) d t$

Step 5

Simplify the expression.

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Step 5.1

Simplify.

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Step 5.1.1

Combine $d$ and $\frac{1}{6}$ .

$- \frac{d}{6} \int \frac{1}{u^{4}} d u t$

Step 5.1.2

Combine $t$ and $\frac{d}{6}$ .

$- \frac{t d}{6} \int \frac{1}{u^{4}} d u$

Step 5.2

Apply basic rules of exponents.

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Step 5.2.1

Move $u^{4}$ out of the denominator by raising it to the $- 1$ power.

$- \frac{t d}{6} \int {(u^{4})}^{- 1} d u$

Step 5.2.2

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

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Step 5.2.2.1

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

$- \frac{t d}{6} \int u^{4 \cdot - 1} d u$

Step 5.2.2.2

Multiply $4$ by $- 1$ .

$- \frac{t d}{6} \int u^{- 4} d u$

Step 6

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

$- \frac{t d}{6} (- \frac{1}{3} u^{- 3} + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

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Step 7.1.1

Combine $u^{- 3}$ and $\frac{1}{3}$ .

$- \frac{t d}{6} (- \frac{u^{- 3}}{3} + C)$

Step 7.1.2

Move $u^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{t d}{6} (- \frac{1}{3 u^{3}} + C)$

Step 7.2

Rewrite $- \frac{t d}{6} (- \frac{1}{3 u^{3}} + C)$ as $- \frac{1}{6} t d (- \frac{1}{3 u^{3}}) + C$ .

$- \frac{1}{6} t d (- \frac{1}{3 u^{3}}) + C$

Step 7.3

Simplify.

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Step 7.3.1

Combine $t$ and $\frac{1}{6}$ .

$- \frac{t}{6} d (- \frac{1}{3 u^{3}}) + C$

Step 7.3.2

Combine $d$ and $\frac{t}{6}$ .

$- \frac{d t}{6} (- \frac{1}{3 u^{3}}) + C$

Step 7.3.3

Multiply $- 1$ by $- 1$ .

$1 \frac{d t}{6} \frac{1}{3 u^{3}} + C$

Step 7.3.4

Multiply $\frac{d t}{6}$ by $1$ .

$\frac{d t}{6} \cdot \frac{1}{3 u^{3}} + C$

Step 7.3.5

Multiply $\frac{d t}{6}$ by $\frac{1}{3 u^{3}}$ .

$\frac{d t}{6 (3 u^{3})} + C$

Step 7.3.6

Multiply $3$ by $6$ .

$\frac{d t}{18 u^{3}} + C$

Step 8

Replace all occurrences of $u$ with $1 - 6 t$ .

$\frac{d t}{18 {(1 - 6 t)}^{3}} + C$