Calculus Examples

$2 x^{2} - x + 6 - x^{2} + 3 x - 2$

Step 1

Simplify.

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Step 1.1

Subtract $x^{2}$ from $2 x^{2}$ .

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

Step 1.2

Add $- x$ and $3 x$ .

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

Step 1.3

Subtract $2$ from $6$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

$\frac{1}{3} x^{3} + C + \int 2 x d x + \int 4 d x$

Step 4

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

$\frac{1}{3} x^{3} + C + 2 \int x d x + \int 4 d x$

Step 5

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

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

Step 6

Apply the constant rule.

$\frac{1}{3} x^{3} + C + 2 (\frac{1}{2} x^{2} + C) + 4 x + C$

Step 7

Simplify.

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Step 7.1

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

$\frac{1}{3} x^{3} + C + 2 (\frac{x^{2}}{2} + C) + 4 x + C$

Step 7.2

Simplify.

$\frac{x^{3}}{3} + x^{2} + 4 x + C$

Step 7.3

Reorder terms.

$\frac{1}{3} x^{3} + x^{2} + 4 x + C$