Calculus Examples

$\int (7 + t) \sqrt{t} d t$

Step 1

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

$\int (7 + t) t^{\frac{1}{2}} d t$

Step 2

Expand $(7 + t) t^{\frac{1}{2}}$ .

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Step 2.1

Apply the distributive property.

$\int 7 t^{\frac{1}{2}} + t \cdot t^{\frac{1}{2}} d t$

Step 2.2

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

$\int 7 t^{\frac{1}{2}} + t^{1} t^{\frac{1}{2}} d t$

Step 2.3

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

$\int 7 t^{\frac{1}{2}} + t^{1 + \frac{1}{2}} d t$

Step 2.4

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

$\int 7 t^{\frac{1}{2}} + t^{\frac{2}{2} + \frac{1}{2}} d t$

Step 2.5

Combine the numerators over the common denominator.

$\int 7 t^{\frac{1}{2}} + t^{\frac{2 + 1}{2}} d t$

Step 2.6

Add $2$ and $1$ .

$\int 7 t^{\frac{1}{2}} + t^{\frac{3}{2}} d t$

Step 3

Split the single integral into multiple integrals.

$\int 7 t^{\frac{1}{2}} d t + \int t^{\frac{3}{2}} d t$

Step 4

Since $7$ is constant with respect to $t$ , move $7$ out of the integral.

$7 \int t^{\frac{1}{2}} d t + \int t^{\frac{3}{2}} d t$

Step 5

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

$7 (\frac{2}{3} t^{\frac{3}{2}} + C) + \int t^{\frac{3}{2}} d t$

Step 6

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

$7 (\frac{2}{3} t^{\frac{3}{2}} + C) + \frac{2}{5} t^{\frac{5}{2}} + C$

Step 7

Simplify.

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Step 7.1

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

$7 (\frac{2 t^{\frac{3}{2}}}{3} + C) + \frac{2}{5} t^{\frac{5}{2}} + C$

Step 7.2

Simplify.

$\frac{14 t^{\frac{3}{2}}}{3} + \frac{2}{5} t^{\frac{5}{2}} + C$

Step 7.3

Reorder terms.

$\frac{14}{3} t^{\frac{3}{2}} + \frac{2}{5} t^{\frac{5}{2}} + C$