Find out the answer to "Use properties of the integral and the formulas..."

Oberlaudacu

Answered question

2021-12-09

Use properties of the integral and the formulas in the summary to calculate the integrals.

\int_{0}^{1} (u^{2} - 2 u) d u

Steve Hirano

Beginner2021-12-10Added 34 answers

Step 1
We know that,

\int k x^{n} d x = k \int x^{n} d x

= k (\frac{x^{n + 1}}{n + 1})

Where k is any constant
And

\int_{a}^{b} f (x) d x = F (b) - F (a)

where F(x) is integral of f(x)
Step 2
We have,

\int_{0}^{1} (u^{2} - 2 u) d u = \int_{0}^{1} u^{2} d u - 2 \int_{0}^{1} u d u

= {(\frac{u^{2 + 1}}{2 + 1})}_{0}^{1} - 2 {(\frac{u^{1 + 1}}{1 + 1})}_{0}^{1}

= {(\frac{u^{3}}{3})}_{0}^{1} - 2 {(\frac{u^{2}}{2})}_{0}^{1}

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

= (\frac{1}{3}) - 2 (\frac{1}{2})

= \frac{1}{3} - 1

= \frac{1 - 3}{3}

= \frac{- 2}{3}

Hence, value of integral is

\frac{- 2}{3}

Edward Patten

Beginner2021-12-11Added 38 answers

Given:

\int_{0}^{1} u^{2} - 2 u d u

\int u^{2} - 2 u d u

\int u^{2} d u - \int 2 u d u

\frac{u^{3}}{3} - u^{2}

(\frac{u^{3}}{3} - u^{2}) ∣_{0}^{1}

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

Answer:

- \frac{2}{3}

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