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

Simplify.

Tap for more steps...

Step 2.1

Apply the distributive property.

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

Step 2.2

Reorder $x$ and $1$ .

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

Step 2.3

Multiply $x$ by $1$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 2.8

Add $2$ and $1$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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 5

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 6

Simplify.

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