Calculus Examples

$\int (5 x - 1) \sqrt{x} d x$

Step 1

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

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

Step 2

Expand $(5 x - 1) x^{\frac{1}{2}}$ .

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Step 2.1

Apply the distributive property.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Add $2$ and $1$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

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

$\frac{5 \cdot 2}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + C$

Step 8.2.2

Multiply $5$ by $2$ .

$\frac{10}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + C$

Step 8.2.3

Cancel the common factor of $10$ and $5$ .

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Step 8.2.3.1

Factor $5$ out of $10$ .

$\frac{5 \cdot 2}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + C$

Step 8.2.3.2

Cancel the common factors.

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Step 8.2.3.2.1

Factor $5$ out of $5$ .

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

Step 8.2.3.2.2

Cancel the common factor.

$\frac{5 \cdot 2}{5 \cdot 1} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + C$

Step 8.2.3.2.3

Rewrite the expression.

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

Step 8.2.3.2.4

Divide $2$ by $1$ .

$2 x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + C$