Calculus Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Reorder $x$ and $- 1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Add $2$ and $1$ .

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

Step 2.8

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

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{2}$ and $x^{2}$ .

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

Step 7.2

Simplify.

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