Calculus Examples

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

Step 1

$\int (2 x^{3} + 2 x) u (x) d x$ is an antiderivative of $(2 x^{3} + 2 x) u (x)$ , so by definition $\frac{d}{d x} [\int (2 x^{3} + 2 x) u (x) d x]$ is $(2 x^{3} + 2 x) u (x)$ .

$(2 x^{3} + 2 x) u (x)$

Step 2

Simplify.

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Step 2.1

Apply the distributive property.

$(2 x^{3} u + 2 x u) x$

Step 2.2

Apply the distributive property.

$2 x^{3} u x + 2 x u x$

Step 2.3

Combine terms.

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Step 2.3.1

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

$2 u (x^{3} x^{1}) + 2 x u x$

Step 2.3.2

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

$2 u x^{3 + 1} + 2 x u x$

Step 2.3.3

Add $3$ and $1$ .

$2 u x^{4} + 2 x u x$

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

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

$2 u x^{4} + 2 u x^{1 + 1}$

Step 2.3.7

Add $1$ and $1$ .

$2 u x^{4} + 2 u x^{2}$

Step 2.4

Reorder terms.

$2 x^{4} u + 2 x^{2} u$