Expert Answer to "Evaluate the integral. ∫2x2+7x−3x−2dx"

DofotheroU

Answered question

2021-11-07

Evaluate the integral.

\int \frac{2 x^{2} + 7 x - 3}{x - 2} d x

unessodopunsep

Skilled2021-11-08Added 105 answers

Given
The integral is

\int \frac{2 x^{2} + 7 x - 3}{x - 2} d x

solution

\int \frac{2 x^{2} + 7 x - 3}{x - 2} d x = \int \frac{2 {(x - 2)}^{2} + 15 (x - 2) + 19}{x - 2} d x

let u=x-2; du=dx

\int \frac{2 x^{2} + 7 x - 3}{x - 2} d x = \int \frac{2 u^{2} + 15 u + 19}{u} d u

= \int \frac{2 u^{2}}{u} d u + \int \frac{15 u}{u} d u + \int \frac{19}{u} d u

= 2 \int u d u + 15 \int 1 d u + 19 \int \frac{1}{u} d u

= 2 \cdot \frac{u^{2}}{2} + 15 u + 19 \ln u + C

put u=x-2

\int \frac{2 x^{2} + 7 x - 3}{x - 2} d x = {(x - 2)}^{2} + 15 (x - 2) + 19 \ln (x - 2) + C

= x^{2} - 4 x + 4 + 15 x - 30 + 19 \ln (x - 2) + C

= x^{2} + 11 x - 26 + 19 \ln (x - 2) + C

\int \frac{2 x^{2} + 7 x - 3}{x - 2} d x = x^{2} + 11 x + 19 \ln (x - 2) + C^{'} (C^{'} = C - 26)

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