integral from 0 to 1 of x^2(x^3-2)^3

1) Let's compute the integral:

\int_{0}^{1} x^{2} \cdot {(x^{3} - 2)}^{3} d x

Let

u = x^{3} - 2

. Then

du = 3 x^{2} \cdot dx

, so

x^{2} \cdot dx = \frac{du}{3}

\int x^{2} \cdot {(x^{3} - 2)}^{3} d x = \frac{1}{3} \cdot \int u^{3} d u

An antiderivative of

u^{3}

\frac{u^{4}}{4}

Substitute

u

x^{3} - 2

back into the antiderivative

ⅰ

\int x^{2} \cdot {(x^{3} - 2)}^{3} d x

\frac{{(x^{3} - 2)}^{4}}{12}

▷

\frac{{(x^{3} - 2)}^{4}}{12}

2) Let's evaluate over the interval

[0, 1]

\frac{{(1^{3} - 2)}^{4}}{12} - \frac{{(0^{3} - 2)}^{4}}{12}

ⓘ Parentheses are computed first:

1 - 2

-1

■Adding 0 changes nothing:

0 - 2

-2

■Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

{(-2)}^{4}

(-2) \cdot (-2) \cdot (-2) \cdot (-2)

16

\frac{{(-1)}^{4}}{12} - \frac{16}{12}

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

{(-1)}^{4}

(-1) \cdot (-1) \cdot (-1) \cdot (-1)

1

■We simplify the fraction by dividing the numerator and the denominator by

4

, their greatest common divisor:

- \frac{16}{12}

- \frac{4 \cdot 4}{3 \cdot 4}

- \frac{4}{3}

\frac{1}{12} - \frac{4}{3}

ⓘ To add fractions, put them over the same denominator, then add the numerators. We choose

12

, the least common multiple of

12

and

3

. Multiplying the numerator and the denominator by the same number does not change the fraction:

\frac{1}{12} - \frac{4}{3}

\frac{1}{12} + \frac{-4 \cdot 4}{3 \cdot 4}

\frac{1 - 4 \cdot 4}{12}

\frac{1 - 4 \cdot 4}{12}

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 4

16

\frac{1 - 16}{12}

ⓘ The numerator and the denominator are computed before the division:

1 - 16

-15

- \frac{15}{12}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

3

, their greatest common divisor:

- \frac{15}{12}

- \frac{5 \cdot 3}{4 \cdot 3}

- \frac{5}{4}

▶Integral:

- \frac{5}{4}

3) The numeric value of:

- \frac{5}{4}

is:

▷

-1.25