Calculus Examples

$f (x) = (x + 1) (2 x - 1)$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int (x + 1) (2 x - 1) d x$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\int x (2 x - 1) + 1 (2 x - 1) d x$

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Reorder $x$ and $2$ .

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

Step 3.5

Reorder $x$ and $- 1$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Add $1$ and $1$ .

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

Step 3.10

Multiply $2$ by $1$ .

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

Step 3.11

Multiply $- 1$ by $1$ .

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

Step 3.12

Add $- 1 x$ and $2 x$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Apply the constant rule.

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

Step 10

Simplify.

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Step 10.1

Simplify.

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

Step 10.2

Reorder terms.

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

Step 11

The answer is the antiderivative of the function $f (x) = (x + 1) (2 x - 1)$ .

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