Calculus Examples

$\int \frac{t^{2}}{2} - \frac{2}{t^{2}} d t$

Step 1

Split the single integral into multiple integrals.

$\int \frac{t^{2}}{2} d t + \int - \frac{2}{t^{2}} d t$

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify the expression.

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Step 6.1

Multiply $2$ by $- 1$ .

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

Step 6.2

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

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

Step 6.3

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

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Step 6.3.1

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

$\frac{1}{2} (\frac{1}{3} t^{3} + C) - 2 \int t^{2 \cdot - 1} d t$

Step 6.3.2

Multiply $2$ by $- 1$ .

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

$\frac{1}{2} \cdot \frac{1}{3} t^{3} - 2 (- t^{- 1}) + C$

Step 8.2

Simplify.

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Step 8.2.1

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

$\frac{1}{2 \cdot 3} t^{3} - 2 (- t^{- 1}) + C$

Step 8.2.2

Multiply $2$ by $3$ .

$\frac{1}{6} t^{3} - 2 (- t^{- 1}) + C$

Step 8.2.3

Multiply $- 1$ by $- 2$ .

$\frac{1}{6} t^{3} + 2 t^{- 1} + C$