Calculus Examples

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

Step 1

Split the fraction into multiple fractions.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x^{3}$ and $x$ .

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Step 3.1.1

Factor $x$ out of $4 x^{3}$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

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

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

Step 3.1.2.2

Factor $x$ out of $x^{1}$ .

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

Step 3.1.2.3

Cancel the common factor.

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

Step 3.1.2.4

Rewrite the expression.

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

Step 3.1.2.5

Divide $4 x^{2}$ by $1$ .

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

Step 3.2

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

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

Step 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 7

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

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

Step 8

Simplify the expression.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 8.1.2

Multiply $5$ by $- 1$ .

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

Step 8.2

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

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

Step 8.3

Simplify.

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Step 8.3.1

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 8.3.2

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

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Step 8.3.2.1

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

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Step 8.3.2.1.1

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

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

Step 8.3.2.1.2

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

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

Step 8.3.2.2

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

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

Step 8.3.2.3

Combine the numerators over the common denominator.

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

Step 8.3.2.4

Subtract $1$ from $2$ .

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

Step 8.4

Apply basic rules of exponents.

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Step 8.4.1

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

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

Step 8.4.2

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

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Step 8.4.2.1

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

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

Step 8.4.2.2

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

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

Step 8.4.2.3

Move the negative in front of the fraction.

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

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

Step 10.2

Simplify the expression.

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Step 10.2.1

Multiply $2$ by $- 5$ .

$\frac{4 x^{3}}{3} - 10 x^{\frac{1}{2}} + C$

Step 10.2.2

Reorder terms.

$\frac{4}{3} x^{3} - 10 x^{\frac{1}{2}} + C$