Find-the-x-1-x-2-x-1-dx-

Question Number 32139 by Cheyboy last updated on 20/Mar/18

F i n d t h e

\int \frac{x + 1}{x^{2} + x + 1} d x

Commented by mrW2 last updated on 20/Mar/18

\int \frac{x + 1}{x^{2} + x + 1} d x

= \frac{1}{2} \int \frac{2 x + 1 + 1}{x^{2} + x + 1} d x

= \frac{1}{2} [\int \frac{2 x + 1}{x^{2} + x + 1} d x + \int \frac{1}{x^{2} + x + 1} d x]

= \frac{1}{2} [\int \frac{1}{x^{2} + x + 1} d (x^{2} + x + 1) + \int \frac{1}{x^{2} + x + 1} d x]

= \frac{1}{2} [\ln (x^{2} + x + 1) + \int \frac{1}{{(x + \frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2}} d (x + \frac{1}{2})]

= \frac{1}{2} [\ln (x^{2} + x + 1) + \frac{2}{\sqrt{3}} \tan^{- 1} \frac{2 x + 1}{\sqrt{3}}] + C

= \frac{1}{2} \ln (x^{2} + x + 1) + \frac{1}{\sqrt{3}} \tan^{- 1} \frac{2 x + 1}{\sqrt{3}} + C

Commented by Cheyboy last updated on 20/Mar/18

S i r p l z z c a n u e x p l a i n t h i s p a r t

\int \frac{1}{x^{2} + x + 1} d (x^{2} + x + 1)

Commented by Tinkutara last updated on 20/Mar/18

\frac{d}{d x} (x^{2} + x + 1) = 2 x + 1

d (x^{2} + x + 1) = (2 x + 1) d x

(2 x + 1) d x i s r e p l a c e d b y d (x^{2} + x + 1) .

Commented by Cheyboy last updated on 20/Mar/18

O o h!! i h a v e s e e n i t T h a n k y o u f o r

t h e c l e a r i f i c a t i o n .

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