Calculus Examples

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

Step 1

$\int (x + 1) (x + a) d x$ is an antiderivative of $(x + 1) (x + a)$ , so by definition $\frac{d}{d x} [\int (x + 1) (x + a) d x]$ is $(x + 1) (x + a)$ .

$(x + 1) (x + a)$

Step 2

Simplify.

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Step 2.1

Apply the distributive property.

$(x + 1) x + (x + 1) a$

Step 2.2

Apply the distributive property.

$x \cdot x + 1 x + (x + 1) a$

Step 2.3

Apply the distributive property.

$x \cdot x + 1 x + x a + 1 a$

Step 2.4

Combine terms.

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Step 2.4.1

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

$x^{1} x + 1 x + x a + 1 a$

Step 2.4.2

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

$x^{1} x^{1} + 1 x + x a + 1 a$

Step 2.4.3

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

$x^{1 + 1} + 1 x + x a + 1 a$

Step 2.4.4

Add $1$ and $1$ .

$x^{2} + 1 x + x a + 1 a$

Step 2.4.5

Multiply $x$ by $1$ .

$x^{2} + x + x a + 1 a$

Step 2.4.6

Multiply $a$ by $1$ .

$x^{2} + x + x a + a$

Step 2.5

Reorder terms.

$x^{2} + a + a x + x$