Calculus Examples

$\int \frac{u^{3} + u}{u} d u$

Step 1

Split the fraction into multiple fractions.

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Step 1.1

Factor $u$ out of $u^{3} + u$ .

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Step 1.1.1

Factor $u$ out of $u^{3}$ .

$\int \frac{u \cdot u^{2} + u}{u} d u$

Step 1.1.2

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

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

Step 1.1.3

Factor $u$ out of $u^{1}$ .

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

Step 1.1.4

Factor $u$ out of $u \cdot u^{2} + u \cdot 1$ .

$\int \frac{u (u^{2} + 1)}{u} d u$

Step 1.2

Cancel the common factor of $u$ .

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Step 1.2.1

Cancel the common factor.

$\int \frac{u (u^{2} + 1)}{u} d u$

Step 1.2.2

Divide $u^{2} + 1$ by $1$ .

$\int u^{2} + 1 d u$

Step 2

Split the single integral into multiple integrals.

$\int u^{2} d u + \int d u$

Step 3

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

$\frac{1}{3} u^{3} + C + \int d u$

Step 4

Apply the constant rule.

$\frac{1}{3} u^{3} + C + u + C$

Step 5

Simplify.

$\frac{1}{3} u^{3} + u + C$