Calculus Examples

$\frac{6 + t + t^{2}}{\sqrt{t}}$

Step 1

Write $\frac{6 + t + t^{2}}{\sqrt{t}}$ as a function.

$f (t) = \frac{6 + t + t^{2}}{\sqrt{t}}$

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.

$F (t) = \int \frac{6 + t + t^{2}}{\sqrt{t}} d t$

Step 4

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

$\int \frac{6 + t + t^{2}}{t^{\frac{1}{2}}} d t$

Step 5

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

$\int (6 + t + t^{2}) {(t^{\frac{1}{2}})}^{- 1} d t$

Step 6

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

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Step 6.1

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

$\int (6 + t + t^{2}) t^{\frac{1}{2} \cdot - 1} d t$

Step 6.2

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

$\int (6 + t + t^{2}) t^{\frac{- 1}{2}} d t$

Step 6.3

Move the negative in front of the fraction.

$\int (6 + t + t^{2}) t^{- \frac{1}{2}} d t$

Step 7

Expand $(6 + t + t^{2}) t^{- \frac{1}{2}}$ .

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Step 7.1

Apply the distributive property.

$\int (6 + t) t^{- \frac{1}{2}} + t^{2} t^{- \frac{1}{2}} d t$

Step 7.2

Apply the distributive property.

$\int 6 t^{- \frac{1}{2}} + t \cdot t^{- \frac{1}{2}} + t^{2} t^{- \frac{1}{2}} d t$

Step 7.3

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

$\int 6 t^{- \frac{1}{2}} + t^{1} t^{- \frac{1}{2}} + t^{2} t^{- \frac{1}{2}} d t$

Step 7.4

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

$\int 6 t^{- \frac{1}{2}} + t^{1 - \frac{1}{2}} + t^{2} t^{- \frac{1}{2}} d t$

Step 7.5

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

$\int 6 t^{- \frac{1}{2}} + t^{\frac{2}{2} - \frac{1}{2}} + t^{2} t^{- \frac{1}{2}} d t$

Step 7.6

Combine the numerators over the common denominator.

$\int 6 t^{- \frac{1}{2}} + t^{\frac{2 - 1}{2}} + t^{2} t^{- \frac{1}{2}} d t$

Step 7.7

Subtract $1$ from $2$ .

$\int 6 t^{- \frac{1}{2}} + t^{\frac{1}{2}} + t^{2} t^{- \frac{1}{2}} d t$

Step 7.8

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

$\int 6 t^{- \frac{1}{2}} + t^{\frac{1}{2}} + t^{2 - \frac{1}{2}} d t$

Step 7.9

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

$\int 6 t^{- \frac{1}{2}} + t^{\frac{1}{2}} + t^{2 \cdot \frac{2}{2} - \frac{1}{2}} d t$

Step 7.10

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

$\int 6 t^{- \frac{1}{2}} + t^{\frac{1}{2}} + t^{\frac{2 \cdot 2}{2} - \frac{1}{2}} d t$

Step 7.11

Combine the numerators over the common denominator.

$\int 6 t^{- \frac{1}{2}} + t^{\frac{1}{2}} + t^{\frac{2 \cdot 2 - 1}{2}} d t$

Step 7.12

Simplify the numerator.

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Step 7.12.1

Multiply $2$ by $2$ .

$\int 6 t^{- \frac{1}{2}} + t^{\frac{1}{2}} + t^{\frac{4 - 1}{2}} d t$

Step 7.12.2

Subtract $1$ from $4$ .

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

Step 8

Split the single integral into multiple integrals.

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

Step 9

Since $6$ is constant with respect to $t$ , move $6$ out of the integral.

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

Step 10

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

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

Step 11

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

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

Step 12

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

$6 (2 t^{\frac{1}{2}} + C) + \frac{2}{3} t^{\frac{3}{2}} + C + \frac{2}{5} t^{\frac{5}{2}} + C$

Step 13

Simplify.

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Step 13.1

Simplify.

$12 t^{\frac{1}{2}} + \frac{2 t^{\frac{3}{2}}}{3} + \frac{2}{5} t^{\frac{5}{2}} + C$

Step 13.2

Reorder terms.

$12 t^{\frac{1}{2}} + \frac{2}{3} t^{\frac{3}{2}} + \frac{2}{5} t^{\frac{5}{2}} + C$

Step 14

The answer is the antiderivative of the function $f (t) = \frac{6 + t + t^{2}}{\sqrt{t}}$ .

$F (t) =$ $12 t^{\frac{1}{2}} + \frac{2}{3} t^{\frac{3}{2}} + \frac{2}{5} t^{\frac{5}{2}} + C$