Calculus Examples

$\int (2 \sqrt{t} - t - \frac{9}{t^{2}}) d t$

Step 1

Remove parentheses.

$\int 2 \sqrt{t} - t - \frac{9}{t^{2}} d t$

Step 2

Split the single integral into multiple integrals.

$\int 2 \sqrt{t} d t + \int - t d t + \int - \frac{9}{t^{2}} d t$

Step 3

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

$2 \int \sqrt{t} d t + \int - t d t + \int - \frac{9}{t^{2}} d t$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$2 \int t^{\frac{1}{2}} d t + \int - t d t + \int - \frac{9}{t^{2}} d t$

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Since $9$ is constant with respect to $t$ , move $9$ out of the integral.

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

Step 10

Simplify the expression.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $\frac{2}{3}$ and $t^{\frac{3}{2}}$ .

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

Step 10.1.2

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

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

Step 10.2

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

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

Step 10.3

Simplify.

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Step 10.3.1

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

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Step 10.3.1.1

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

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

Step 10.3.1.2

Multiply $2$ by $- 1$ .

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

Step 10.3.2

Multiply $9$ by $- 1$ .

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

Step 11

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

$2 (\frac{2 t^{\frac{3}{2}}}{3} + C) - (\frac{t^{2}}{2} + C) - 9 (- t^{- 1} + C)$

Step 12

Simplify.

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Step 12.1

Simplify.

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

Step 12.2

Multiply $- 1$ by $- 9$ .

$\frac{4 t^{\frac{3}{2}}}{3} - \frac{t^{2}}{2} + 9 t^{- 1} + C$

Step 13

Reorder terms.

$\frac{4}{3} t^{\frac{3}{2}} - \frac{1}{2} t^{2} + 9 t^{- 1} + C$