Calculus Examples

$\int x (x^{3} + 1) d x$

Step 1

Multiply $x (x^{3} + 1)$ .

$\int x \cdot x^{3} + x \cdot 1 d x$

Step 2

Simplify.

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Step 2.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 2.1.1

Multiply $x$ by $x^{3}$ .

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Step 2.1.1.1

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

$\int x^{1} x^{3} + x \cdot 1 d x$

Step 2.1.1.2

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

$\int x^{1 + 3} + x \cdot 1 d x$

Step 2.1.2

Add $1$ and $3$ .

$\int x^{4} + x \cdot 1 d x$

Step 2.2

Multiply $x$ by $1$ .

$\int x^{4} + x d x$

Step 3

Split the single integral into multiple integrals.

$\int x^{4} d x + \int x d x$

Step 4

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

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

Step 5

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

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

Step 6

Simplify.

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