Calculus Examples

$\int (2 x^{3} + 2 x) u (x) d x = x^{6} + 2 x^{4} + x^{2} + C$

Step 1

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

$u \int (2 x^{3} + 2 x) (x) d x$

Step 2

Simplify.

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Step 2.1

Apply the distributive property.

$u \int 2 x^{3} x + 2 x \cdot x d x$

Step 2.2

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

$u \int 2 (x^{3} x^{1}) + 2 x \cdot x d x$

Step 2.3

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

$u \int 2 x^{3 + 1} + 2 x \cdot x d x$

Step 2.4

Add $3$ and $1$ .

$u \int 2 x^{4} + 2 x \cdot x d x$

Step 2.5

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

$u \int 2 x^{4} + 2 (x^{1} x) d x$

Step 2.6

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

$u \int 2 x^{4} + 2 (x^{1} x^{1}) d x$

Step 2.7

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

$u \int 2 x^{4} + 2 x^{1 + 1} d x$

Step 2.8

Add $1$ and $1$ .

$u \int 2 x^{4} + 2 x^{2} d x$

Step 3

Split the single integral into multiple integrals.

$u (\int 2 x^{4} d x + \int 2 x^{2} d x)$

Step 4

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

$u (2 \int x^{4} d x + \int 2 x^{2} d x)$

Step 5

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

$u (2 (\frac{1}{5} x^{5} + C) + \int 2 x^{2} d x)$

Step 6

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

$u (2 (\frac{1}{5} x^{5} + C) + 2 \int x^{2} d x)$

Step 7

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

$u (2 (\frac{1}{5} x^{5} + C) + 2 (\frac{1}{3} x^{3} + C))$

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Reorder terms.

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