Calculus Examples

$\int (2 x^{3} - 4 x + 3 x) d x$

Step 1

Remove parentheses.

$\int 2 x^{3} - 4 x + 3 x d x$

Step 2

Add $- 4 x$ and $3 x$ .

$\int 2 x^{3} - x d x$

Step 3

Split the single integral into multiple integrals.

$\int 2 x^{3} d x + \int - x d x$

Step 4

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

$2 \int x^{3} d x + \int - x d x$

Step 5

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$2 (\frac{1}{4} x^{4} + C) + \int - x d x$

Step 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$2 (\frac{1}{4} x^{4} + C) - \int x d x$

Step 7

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

$2 (\frac{1}{4} x^{4} + C) - (\frac{1}{2} x^{2} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$2 (\frac{1}{4}) x^{4} - \frac{1}{2} x^{2} + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $2$ and $\frac{1}{4}$ .

$\frac{2}{4} x^{4} - \frac{1}{2} x^{2} + C$

Step 8.2.2

Cancel the common factor of $2$ and $4$ .

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Step 8.2.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4} x^{4} - \frac{1}{2} x^{2} + C$

Step 8.2.2.2

Cancel the common factors.

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Step 8.2.2.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2} x^{4} - \frac{1}{2} x^{2} + C$

Step 8.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2} x^{4} - \frac{1}{2} x^{2} + C$

Step 8.2.2.2.3

Rewrite the expression.

$\frac{1}{2} x^{4} - \frac{1}{2} x^{2} + C$