integral from 2 to 5 of (4x-2) — step-by-step solution

1) Let's compute the integral:

\int_{2}^{5} (4 x - 2) d x

ⅰ

\int (4 x - 2) d x

\int 4 x d x - \int 2 d x

\int 4 x d x - \int 2 d x

ⓘ A constant factor stays in front of the integral. An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

\int 4 x d x

4 \cdot \frac{x^{2}}{2}

■The derivative of

- 2 x

-2

, so

- 2 x

is an antiderivative of

-2

- \int 2 d x

- 2 x

4 \cdot \frac{x^{2}}{2} - 2 x

ⓘ Multiplying a fraction by a number multiplies its numerator:

4 \cdot \frac{x^{2}}{2}

\frac{4 x^{2}}{2}

■A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{4 x^{2}}{2}

2 x^{2}

2 x^{2} - 2 x

This expression can be factorized into:

x \cdot (2 x - 2)

Let's factorize in

ℝ

(2 x - 2)

We factor out

2

, the greatest common divisor of the coefficients:

2 x - 2

2 \cdot (x - 1)

Factor identified:

x - 1

▷

2 x \cdot (x - 1)

2) Let's evaluate for the interval

[2, 5]

2 \cdot 5 \cdot (5 - 1) - 2 \cdot 2 \cdot (2 - 1)

ⓘ Multiplications are done before additions and subtractions:

2 \cdot 5

10

2 \cdot 2

4

■Parentheses are computed first:

5 - 1

4

2 - 1

1

■Multiplying by 1 changes nothing:

4 \cdot 1

4

10 \cdot 4 - 4

ⓘ Multiplications are done before additions and subtractions:

10 \cdot 4

40

40 - 4

ⓘ Start from

40

and move back

4

on the number line:

40 - 4

36

▶Integral:

36