Calculus Examples

$\int \frac{4 + 5 x^{\frac{3}{2}}}{\sqrt{x}} d x$

Step 1

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

$\int \frac{4 + 5 x^{\frac{3}{2}}}{x^{\frac{1}{2}}} d x$

Step 2

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

$\int (4 + 5 x^{\frac{3}{2}}) {(x^{\frac{1}{2}})}^{- 1} d x$

Step 3

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

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Step 3.1

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

$\int (4 + 5 x^{\frac{3}{2}}) x^{\frac{1}{2} \cdot - 1} d x$

Step 3.2

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

$\int (4 + 5 x^{\frac{3}{2}}) x^{\frac{- 1}{2}} d x$

Step 3.3

Move the negative in front of the fraction.

$\int (4 + 5 x^{\frac{3}{2}}) x^{- \frac{1}{2}} d x$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\int 4 x^{- \frac{1}{2}} + 5 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.2

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

$\int 4 x^{- \frac{1}{2}} + 5 x^{\frac{3}{2} - \frac{1}{2}} d x$

Step 4.3

Combine the numerators over the common denominator.

$\int 4 x^{- \frac{1}{2}} + 5 x^{\frac{3 - 1}{2}} d x$

Step 4.4

Subtract $1$ from $3$ .

$\int 4 x^{- \frac{1}{2}} + 5 x^{\frac{2}{2}} d x$

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Cancel the common factor.

$\int 4 x^{- \frac{1}{2}} + 5 x^{\frac{2}{2}} d x$

Step 4.5.2

Rewrite the expression.

$\int 4 x^{- \frac{1}{2}} + 5 x^{1} d x$

Step 4.6

Simplify.

$\int 4 x^{- \frac{1}{2}} + 5 x d x$

Step 4.7

Reorder $4 x^{- \frac{1}{2}}$ and $5 x$ .

$\int 5 x + 4 x^{- \frac{1}{2}} d x$

Step 5

Split the single integral into multiple integrals.

$\int 5 x d x + \int 4 x^{- \frac{1}{2}} d x$

Step 6

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int x d x + \int 4 x^{- \frac{1}{2}} d x$

Step 7

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

$5 (\frac{1}{2} x^{2} + C) + \int 4 x^{- \frac{1}{2}} d x$

Step 8

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$5 (\frac{1}{2} x^{2} + C) + 4 \int x^{- \frac{1}{2}} d x$

Step 9

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

$5 (\frac{1}{2} x^{2} + C) + 4 (2 x^{\frac{1}{2}} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$5 (\frac{1}{2}) x^{2} + 4 \cdot 2 x^{\frac{1}{2}} + C$

Step 10.2

Simplify.

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Step 10.2.1

Combine $5$ and $\frac{1}{2}$ .

$\frac{5}{2} x^{2} + 4 \cdot 2 x^{\frac{1}{2}} + C$

Step 10.2.2

Multiply $4$ by $2$ .

$\frac{5}{2} x^{2} + 8 x^{\frac{1}{2}} + C$