Calculus Examples

$\int x^{2} - 2 x + 5 d x$

Step 1

Split the single integral into multiple integrals.

$\int x^{2} d x + \int - 2 x d x + \int 5 d x$

Step 2

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

$\frac{1}{3} x^{3} + C + \int - 2 x d x + \int 5 d x$

Step 3

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

$\frac{1}{3} x^{3} + C - 2 \int x d x + \int 5 d x$

Step 4

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

$\frac{1}{3} x^{3} + C - 2 (\frac{1}{2} x^{2} + C) + \int 5 d x$

Step 5

Apply the constant rule.

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Simplify.

$\frac{x^{3}}{3} - x^{2} + 5 x + C$

Step 6.3

Reorder terms.

$\frac{1}{3} x^{3} - x^{2} + 5 x + C$