Calculus Examples

$g (t) = \frac{1 + t + t^{2}}{\sqrt{t}}$

Step 1

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

$G (t) = \int g (t) d t$

Step 2

Set up the integral to solve.

$G (t) = \int \frac{1 + t + t^{2}}{\sqrt{t}} d t$

Step 3

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

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Move the negative in front of the fraction.

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

Step 6

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

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $t^{- \frac{1}{2}}$ by $1$ .

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

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

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

Step 6.7

Combine the numerators over the common denominator.

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

Step 6.8

Subtract $1$ from $2$ .

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

Step 6.9

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

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

Step 6.10

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.11

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

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

Step 6.12

Combine the numerators over the common denominator.

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

Step 6.13

Simplify the numerator.

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Step 6.13.1

Multiply $2$ by $2$ .

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

Step 6.13.2

Subtract $1$ from $4$ .

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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

Step 12

The answer is the antiderivative of the function $g (t) = \frac{1 + t + t^{2}}{\sqrt{t}}$ .

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