Calculus Examples

$\int (x - 1) (x + a) d x$

Step 1

Expand $(x - 1) (x + a)$ .

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Step 1.1

Apply the distributive property.

$\int x (x + a) - 1 (x + a) d x$

Step 1.2

Apply the distributive property.

$\int x \cdot x + x a - 1 (x + a) d x$

Step 1.3

Apply the distributive property.

$\int x \cdot x + x a - 1 x - 1 a d x$

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

Add $1$ and $1$ .

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

Step 1.8

Reorder $- 1 x$ and $- 1 a$ .

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

Step 1.9

Move $x a$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Apply the constant rule.

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

Step 5

Simplify.

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Step 5.1

Rewrite $- 1 a$ as $- a$ .

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

Step 5.2

Rewrite $- 1 x$ as $- x$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

Simplify.

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

Step 12

Reorder terms.

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