Calculus Examples

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

Step 1

Factor $\sqrt{t}$ out of $t \sqrt{t} + \sqrt{t}$ .

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Step 1.1

Factor $\sqrt{t}$ out of $t \sqrt{t}$ .

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

Step 1.2

Multiply by $1$ .

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

Step 1.3

Factor $\sqrt{t}$ out of $\sqrt{t} t + \sqrt{t} \cdot 1$ .

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

Step 2

Simplify the expression.

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Step 2.1

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

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

Step 2.2

Simplify.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

Multiply $2$ by $- 1$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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Step 2.2.3.1

Move $t^{- 2}$ .

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 2.2.3.4

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

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

Step 2.2.3.5

Combine the numerators over the common denominator.

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

Step 2.2.3.6

Simplify the numerator.

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Step 2.2.3.6.1

Multiply $- 2$ by $2$ .

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

Step 2.2.3.6.2

Add $- 4$ and $1$ .

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

Step 2.2.3.7

Move the negative in front of the fraction.

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

Step 3

Expand $t^{- \frac{3}{2}} (t + 1)$ .

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Step 3.1

Apply the distributive property.

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

Step 3.2

Reorder $t^{- \frac{3}{2}}$ and $1$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Combine the numerators over the common denominator.

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

Step 3.7

Add $- 3$ and $2$ .

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

Step 3.8

Multiply $t^{- \frac{3}{2}}$ by $1$ .

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

Step 4

Move the negative in front of the fraction.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

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

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

Step 8.2.2

Move the negative in front of the fraction.

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