Calculus Examples

$\int \frac{2 t^{3} + 1}{{(t^{4} + 2 t)}^{3}} d t$

Step 1

Let $u = t^{4} + 2 t$ . Then $d u = (4 t^{3} + 2) d t$ , so $\frac{1}{2} d u = (2 t^{3} + 1) d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = t^{4} + 2 t$ . Find $\frac{d u}{d t}$ .

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Step 1.1.1

Differentiate $t^{4} + 2 t$ .

$\frac{d}{d t} [t^{4} + 2 t]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $t^{4} + 2 t$ with respect to $t$ is $\frac{d}{d t} [t^{4}] + \frac{d}{d t} [2 t]$ .

$\frac{d}{d t} [t^{4}] + \frac{d}{d t} [2 t]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 4$ .

$4 t^{3} + \frac{d}{d t} [2 t]$

Step 1.1.3

Evaluate $\frac{d}{d t} [2 t]$ .

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Step 1.1.3.1

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$4 t^{3} + 2 \frac{d}{d t} [t]$

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 $n = 1$ .

$4 t^{3} + 2 \cdot 1$

Step 1.1.3.3

Multiply $2$ by $1$ .

$4 t^{3} + 2$

Step 1.2

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

$\int \frac{1}{u^{3}} \cdot \frac{1}{2} d u$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{1}{u^{3}}$ by $\frac{1}{2}$ .

$\int \frac{1}{u^{3} \cdot 2} d u$

Step 2.2

Move $2$ to the left of $u^{3}$ .

$\int \frac{1}{2 u^{3}} d u$

Step 3

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

$\frac{1}{2} \int \frac{1}{u^{3}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $u^{3}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{2} \int {(u^{3})}^{- 1} d u$

Step 4.2

Multiply the exponents in ${(u^{3})}^{- 1}$ .

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Step 4.2.1

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

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

Step 4.2.2

Multiply $3$ by $- 1$ .

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

Step 5

By the Power Rule, the integral of $u^{- 3}$ with respect to $u$ is $- \frac{1}{2} u^{- 2}$ .

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

Step 6

Simplify.

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Step 6.1

Rewrite $\frac{1}{2} (- \frac{1}{2} u^{- 2} + C)$ as $\frac{1}{2} (- \frac{1}{2} \cdot \frac{1}{u^{2}}) + C$ .

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

Step 6.2

Simplify.

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Step 6.2.1

Multiply $\frac{1}{u^{2}}$ by $\frac{1}{2}$ .

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

Step 6.2.2

Move $2$ to the left of $u^{2}$ .

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

Step 6.2.3

Multiply $\frac{1}{2}$ by $\frac{1}{2 u^{2}}$ .

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

Step 6.2.4

Multiply $2$ by $2$ .

$- \frac{1}{4 u^{2}} + C$

Step 7

Replace all occurrences of $u$ with $t^{4} + 2 t$ .

$- \frac{1}{4 {(t^{4} + 2 t)}^{2}} + C$