Calculus Examples

$\int 2 x^{2} - 6 x + \frac{4}{x} - 3 d x$

Step 1

Split the single integral into multiple integrals.

$\int 2 x^{2} d x + \int - 6 x d x + \int \frac{4}{x} d x + \int - 3 d x$

Step 2

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

$2 \int x^{2} d x + \int - 6 x d x + \int \frac{4}{x} d x + \int - 3 d x$

Step 3

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

$2 (\frac{1}{3} x^{3} + C) + \int - 6 x d x + \int \frac{4}{x} d x + \int - 3 d x$

Step 4

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$2 (\frac{1}{3} x^{3} + C) - 6 \int x d x + \int \frac{4}{x} d x + \int - 3 d x$

Step 5

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

$2 (\frac{1}{3} x^{3} + C) - 6 (\frac{1}{2} x^{2} + C) + \int \frac{4}{x} d x + \int - 3 d x$

Step 6

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$2 (\frac{1}{3} x^{3} + C) - 6 (\frac{1}{2} x^{2} + C) + 4 \int \frac{1}{x} d x + \int - 3 d x$

Step 7

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$2 (\frac{1}{3} x^{3} + C) - 6 (\frac{1}{2} x^{2} + C) + 4 (\ln (| x |) + C) + \int - 3 d x$

Step 8

Apply the constant rule.

$2 (\frac{1}{3} x^{3} + C) - 6 (\frac{1}{2} x^{2} + C) + 4 (\ln (| x |) + C) - 3 x + C$

Step 9

Simplify.

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Step 9.1

Simplify.

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Step 9.1.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$2 (\frac{x^{3}}{3} + C) - 6 (\frac{1}{2} x^{2} + C) + 4 (\ln (| x |) + C) - 3 x + C$

Step 9.1.2

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

$2 (\frac{x^{3}}{3} + C) - 6 (\frac{x^{2}}{2} + C) + 4 (\ln (| x |) + C) - 3 x + C$

Step 9.2

Simplify.

$\frac{2 x^{3}}{3} - 3 x^{2} + 4 \ln (| x |) - 3 x + C$

Step 9.3

Reorder terms.

$\frac{2}{3} x^{3} - 3 x^{2} + 4 \ln (| x |) - 3 x + C$