Calculus Examples

$\int u^{6} v (2 + u^{7}) d u$

Step 1

Since $v$ is constant with respect to $u$ , move $v$ out of the integral.

$v \int u^{6} (2 + u^{7}) d u$

Step 2

Expand $u^{6} (2 + u^{7})$ .

Tap for more steps...

Step 2.1

Apply the distributive property.

$v \int u^{6} \cdot 2 + u^{6} u^{7} d u$

Step 2.2

Reorder $u^{6}$ and $2$ .

$v \int 2 \cdot u^{6} + u^{6} u^{7} d u$

Step 2.3

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

$v \int 2 u^{6} + u^{6 + 7} d u$

Step 2.4

Add $6$ and $7$ .

$v \int 2 u^{6} + u^{13} d u$

Step 2.5

Reorder $2 u^{6}$ and $u^{13}$ .

$v \int u^{13} + 2 u^{6} d u$

Step 3

Split the single integral into multiple integrals.

$v (\int u^{13} d u + \int 2 u^{6} d u)$

Step 4

By the Power Rule, the integral of $u^{13}$ with respect to $u$ is $\frac{1}{14} u^{14}$ .

$v (\frac{1}{14} u^{14} + C + \int 2 u^{6} d u)$

Step 5

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$v (\frac{1}{14} u^{14} + C + 2 \int u^{6} d u)$

Step 6

By the Power Rule, the integral of $u^{6}$ with respect to $u$ is $\frac{1}{7} u^{7}$ .

$v (\frac{1}{14} u^{14} + C + 2 (\frac{1}{7} u^{7} + C))$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Simplify.

$v (\frac{u^{14}}{14} + \frac{2 u^{7}}{7}) + C$

Step 7.2

Reorder terms.

$v (\frac{1}{14} u^{14} + \frac{2}{7} u^{7}) + C$