help@askiitians.com

+91-735-322-1155

Integral Calculus> integration prev doubt...

question mark

Plz solve this integration..............

∫ ((√((x)^2+x+1))/(x+1)) dx

Sir in this,

x²+x+1 = (x+1)² - x

where x+1 is replaced by t

then we get

((v((t)^2-(t-1)))/t) dt right; then how to proceed sir.....

where 'v' represents sq.rt

Moni RS , 16 Years ago

Grade 11

1 Answers

Jitender Singh

Last Activity: 11 Years ago

Ans:

$I = \int \frac{\sqrt{x^{2}+x+1}}{x+1}dx$

$I = \int \frac{\sqrt{(x+\frac{1}{2})^{2}+\frac{3}{4}}}{x+1}dx$

$u = x + \frac{1}{2}$

$du = dx$

$I = \int \frac{\sqrt{(u)^{2}+\frac{3}{4}}}{u+\frac{1}{2}}du$

$u = \frac{\sqrt{3}}{2}tan(s)$

$du = \frac{\sqrt{3}}{2}sec^{2}(s).ds$

$I = \frac{3}{4}\int \frac{sec^{3}(s)}{\frac{\sqrt{3}}{2}tan(s)+\frac{1}{2}}ds$

$t = tan (\frac{s}{2})$

$dt = \frac{1}{2}sec^{2} (\frac{s}{2}).ds$

$I = 3\int \frac{(p^{2}+1)^{2}}{(p^{2}-1)^{2}.(-p^{2}+2\sqrt{3}p+1)}dp$

Use simply partial fraction rule here, you will get

$I = \sqrt{x^{2}+x+1}-log(2 \sqrt{x^{2}+x+1}-(x-1))+log(x+1)-\frac{1}{2}sinh^{-1}(\frac{2x+1}{\sqrt{3}}) + c$

Thanks & Regards

Jitender Singh

IIT Delhi

askIITians Faculty

LIVE ONLINE CLASSES

Prepraring for the competition made easy just by live online class.

Full Live Access

Study Material

Live Doubts Solving

Daily Class Assignments

The area of a square field is 640000cm².what is its area in hectares

integral calculus

1 Answer Available

Last Activity: 2 Years ago

Please solve this integration. I am facing difficulty in it
How to solve this. Please solve anyone. Humble request

integral calculus

1 Answer Available

Last Activity: 3 Years ago

Using laplace transform, solve d²y/dt²+ dy/dt = t² +2t given that y=4 and y’=-2 and t=0

integral calculus

1 Answer Available

Last Activity: 3 Years ago

If 4th term of an A.p is zero, show that the 8th term is double the 6th term

integral calculus

1 Answer Available

Last Activity: 3 Years ago

View all Questions