Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Add $2$ and $1$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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