Calculus Examples

$(10 t^{- 3} + 12 t^{- 9} + 4 t^{3}) d t$

Step 1

Remove parentheses.

$\int 10 t^{- 3} + 12 t^{- 9} + 4 t^{3} d t$

Step 2

Split the single integral into multiple integrals.

$\int 10 t^{- 3} d t + \int 12 t^{- 9} d t + \int 4 t^{3} d t$

Step 3

Since $10$ is constant with respect to $t$ , move $10$ out of the integral.

$10 \int t^{- 3} d t + \int 12 t^{- 9} d t + \int 4 t^{3} d t$

Step 4

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

$10 (- \frac{1}{2} t^{- 2} + C) + \int 12 t^{- 9} d t + \int 4 t^{3} d t$

Step 5

Simplify.

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Step 5.1

Combine $t^{- 2}$ and $\frac{1}{2}$ .

$10 (- \frac{t^{- 2}}{2} + C) + \int 12 t^{- 9} d t + \int 4 t^{3} d t$

Step 5.2

Move $t^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$10 (- \frac{1}{2 t^{2}} + C) + \int 12 t^{- 9} d t + \int 4 t^{3} d t$

Step 6

Since $12$ is constant with respect to $t$ , move $12$ out of the integral.

$10 (- \frac{1}{2 t^{2}} + C) + 12 \int t^{- 9} d t + \int 4 t^{3} d t$

Step 7

By the Power Rule, the integral of $t^{- 9}$ with respect to $t$ is $- \frac{1}{8} t^{- 8}$ .

$10 (- \frac{1}{2 t^{2}} + C) + 12 (- \frac{1}{8} t^{- 8} + C) + \int 4 t^{3} d t$

Step 8

Simplify.

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Step 8.1

Combine $t^{- 8}$ and $\frac{1}{8}$ .

$10 (- \frac{1}{2 t^{2}} + C) + 12 (- \frac{t^{- 8}}{8} + C) + \int 4 t^{3} d t$

Step 8.2

Move $t^{- 8}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$10 (- \frac{1}{2 t^{2}} + C) + 12 (- \frac{1}{8 t^{8}} + C) + \int 4 t^{3} d t$

Step 9

Since $4$ is constant with respect to $t$ , move $4$ out of the integral.

$10 (- \frac{1}{2 t^{2}} + C) + 12 (- \frac{1}{8 t^{8}} + C) + 4 \int t^{3} d t$

Step 10

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

$10 (- \frac{1}{2 t^{2}} + C) + 12 (- \frac{1}{8 t^{8}} + C) + 4 (\frac{1}{4} t^{4} + C)$

Step 11

Simplify.

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Step 11.1

Simplify.

$- \frac{5}{t^{2}} - \frac{3}{2 t^{8}} + 4 (\frac{1}{4} t^{4}) + C$

Step 11.2

Simplify.

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Step 11.2.1

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

$- \frac{5}{t^{2}} - \frac{3}{2 t^{8}} + 4 \frac{t^{4}}{4} + C$

Step 11.2.2

Combine $4$ and $\frac{t^{4}}{4}$ .

$- \frac{5}{t^{2}} - \frac{3}{2 t^{8}} + \frac{4 t^{4}}{4} + C$

Step 11.2.3

Cancel the common factor of $4$ .

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Step 11.2.3.1

Cancel the common factor.

$- \frac{5}{t^{2}} - \frac{3}{2 t^{8}} + \frac{4 t^{4}}{4} + C$

Step 11.2.3.2

Divide $t^{4}$ by $1$ .

$- \frac{5}{t^{2}} - \frac{3}{2 t^{8}} + t^{4} + C$