Calculus Examples

$\int t (2 t - 3 \sqrt{t}) d t$

Step 1

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

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

Step 2

Simplify.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Reorder $t$ and $2$ .

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

Step 2.3

Reorder $t$ and $- 3$ .

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

Step 2.4

Raise $t$ to the power of $1$ .

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

Step 2.5

Raise $t$ to the power of $1$ .

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

Step 2.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.7

Add $1$ and $1$ .

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

Step 2.8

Raise $t$ to the power of $1$ .

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

Step 2.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.10

Write $1$ as a fraction with a common denominator.

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

Step 2.11

Combine the numerators over the common denominator.

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

Step 2.12

Add $2$ and $1$ .

$\int 2 t^{2} - 3 t^{\frac{3}{2}} d t$

Step 3

Split the single integral into multiple integrals.

$\int 2 t^{2} d t + \int - 3 t^{\frac{3}{2}} d t$

Step 4

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

$2 \int t^{2} d t + \int - 3 t^{\frac{3}{2}} d t$

Step 5

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

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

Step 6

Since $- 3$ is constant with respect to $t$ , move $- 3$ out of the integral.

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

Step 7

By the Power Rule, the integral of $t^{\frac{3}{2}}$ with respect to $t$ is $\frac{2}{5} t^{\frac{5}{2}}$ .

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

Combine $2$ and $\frac{1}{3}$ .

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

Step 8.2.2

Combine $- 3$ and $\frac{2}{5}$ .

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

Step 8.2.3

Multiply $- 3$ by $2$ .

$\frac{2}{3} t^{3} + \frac{- 6}{5} t^{\frac{5}{2}} + C$

Step 8.2.4

Move the negative in front of the fraction.

$\frac{2}{3} t^{3} - \frac{6}{5} t^{\frac{5}{2}} + C$