Evaluate the definite integral. ∫0−2(5x2−4x+2)dx

Question

Evaluate the definite integral.  ∫0−2(5x2−4x+2)dx

Anot1954 · Accepted Answer

Step 1
Here, we use the result of integration that the integral of

x^{n} i s x^{\frac{n + 1}{n + 1}}

. and the solve the definite integral as shown on board.

\int_{0}^{- 2} (5 x^{2} - 4 x + 2) d x = {| 5 \frac{x^{3}}{3} - 2 x^{2} + 2 x |}_{0}^{- 2}

= (5 \frac{{(- 2)}^{3}}{3} - 2 {(- 2)}^{2} + 2 (- 2)) - (5 \frac{0^{3}}{3} - 2 {(0)}^{2} + 2 (0))

= - \frac{40}{3} - 8 - 4 - 0 = - \frac{76}{3}

Step 2
Ans: -76/3

Nancy Johnson · Accepted Answer

Step 1: If f(x) is a continuous function from a to b, and if F(x) is its integral, then:

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

Step 2: In this case,

f (x) = (5 x^{2} - 4 x + 2)

. Find its integral.

\frac{5 x^{3}}{3} - 2 x^{2} + 2 x ∣_{0}^{- 2}

Step 3: Since

F (x) ∣_{a}^{b} = F (b) - F (a)

, expand the above into F(−2)−F(0):

(\frac{5 {(- 2)}^{3}}{3} - 2 {(- 2)}^{2} + 2 \times - 2) - (\frac{5 \times 0^{3}}{3} - 2 \times 0^{2} + 2 \times 0)

Step 4: Simplify.

= - \frac{76}{3}

Evaluate the definite integral. \int_{0}^{-2}(5x^{2}-4x+2)dx