Calculus Examples

$\int \frac{5}{t^{2}} + 4 t^{2} d t$

Step 1

Split the single integral into multiple integrals.

$\int \frac{5}{t^{2}} d t + \int 4 t^{2} d t$

Step 2

Since $5$ is constant with respect to $t$ , move $5$ out of the integral.

$5 \int \frac{1}{t^{2}} d t + \int 4 t^{2} d t$

Step 3

Apply basic rules of exponents.

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Step 3.1

Move $t^{2}$ out of the denominator by raising it to the $- 1$ power.

$5 \int {(t^{2})}^{- 1} d t + \int 4 t^{2} d t$

Step 3.2

Multiply the exponents in ${(t^{2})}^{- 1}$ .

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Step 3.2.1

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

$5 \int t^{2 \cdot - 1} d t + \int 4 t^{2} d t$

Step 3.2.2

Multiply $2$ by $- 1$ .

$5 \int t^{- 2} d t + \int 4 t^{2} d t$

Step 4

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

$5 (- t^{- 1} + C) + \int 4 t^{2} d t$

Step 5

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

$5 (- t^{- 1} + C) + 4 \int t^{2} d t$

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Simplify.

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

Step 7.2

Simplify.

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Step 7.2.1

Multiply $- 1$ by $5$ .

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

Step 7.2.2

Combine $- 5$ and $\frac{1}{t}$ .

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

Step 7.2.3

Move the negative in front of the fraction.

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

Step 7.2.4

Combine $\frac{1}{3}$ and $t^{3}$ .

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

Step 7.2.5

Combine $4$ and $\frac{t^{3}}{3}$ .

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

Step 7.3

Reorder terms.

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