Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 1.4

Reorder $x^{2}$ and $2$ .

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

Step 1.5

Reorder $x^{2}$ and $- 1$ .

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

Step 1.6

Multiply $2$ by $1$ .

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

Step 1.7

Multiply $- 1$ by $1$ .

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

Step 1.8

Factor out negative.

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

Step 1.9

Raise $x$ to the power of $1$ .

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

Step 1.10

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.11

Add $2$ and $1$ .

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

Step 1.12

Reorder $2$ and $- x$ .

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

Step 1.13

Reorder $2 x^{2}$ and $- x^{3}$ .

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

Step 1.14

Move $2$ .

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

Step 1.15

Move $- x$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Apply the constant rule.

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

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $\frac{1}{4}$ and $x^{4}$ .

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

Step 10.1.2

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

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

Step 10.1.3

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

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

Step 10.2

Simplify.

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

Step 11

Reorder terms.

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