Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Simplify.

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

Step 6.2

Simplify.

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Step 6.2.1

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

$\frac{2}{2} x^{2} - \frac{1}{3} x^{3} + C$

Step 6.2.2

Cancel the common factor of $2$ .

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Step 6.2.2.1

Cancel the common factor.

$\frac{2}{2} x^{2} - \frac{1}{3} x^{3} + C$

Step 6.2.2.2

Rewrite the expression.

$1 x^{2} - \frac{1}{3} x^{3} + C$

Step 6.2.3

Multiply $x^{2}$ by $1$ .

$x^{2} - \frac{1}{3} x^{3} + C$