Calculus Examples

$3 \cdot x^{2} - 2 \cdot x - 1$

Step 1

Split the single integral into multiple integrals.

$\int 3 \cdot x^{2} d x + \int - 2 \cdot x d x + \int - 1 d x$

Step 2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int x^{2} d x + \int - 2 \cdot x d x + \int - 1 d x$

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Apply the constant rule.

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

Step 7

Simplify.

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Step 7.1

Simplify.

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.2

Simplify.

$x^{3} - x^{2} - x + C$