Calculus Examples

$(2 - t) \sqrt{t}$

Step 1

Write $(2 - t) \sqrt{t}$ as a function.

$f (t) = (2 - t) \sqrt{t}$

Step 2

The function $F (t)$ can be found by finding the indefinite integral of the derivative $f (t)$ .

$F (t) = \int f (t) d t$

Step 3

Set up the integral to solve.

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

Step 4

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

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

Step 5

Expand $(2 - t) t^{\frac{1}{2}}$ .

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Step 5.1

Apply the distributive property.

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

Step 5.2

Factor out negative.

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

Combine the numerators over the common denominator.

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

Step 5.7

Add $2$ and $1$ .

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Simplify.

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

Step 11.2

Simplify.

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Step 11.2.1

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

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

Step 11.2.2

Multiply $2$ by $2$ .

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

Step 12

The answer is the antiderivative of the function $f (t) = (2 - t) \sqrt{t}$ .

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