Q

Solve! - Integral Calculus - JEE Main-2

The integral is equal to:

(where C is a constant of integration)

• Option 1)

• Option 2)

• Option 3)

• Option 4)

87 Views

As learnt in concept

Integration by substitution -

The functions when on substitution of the variable of integration to some quantity gives any one of standard formulas.

- wherein

 Since $\int f(x)dx=\int f(t)dt=\int f(\theta )d\theta$ all variables must be converted into single variable ,$\left ( t\, or\ \theta \right )$

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

$Put \:x+1=t^2; dx=2tdt$

$I=\int \frac{(t^2 +1)2tdt}{[(t^2-1)^2+3(t^2-1)+3]\sqrt{t^2}}$

$I=2 \int \frac{(t^2 +1)tdt}{(t^4 + t^2+1)}$

$I=2 \int \frac{1+1/t^2}{t^2+1+1/t^2} dt$

$I= \frac{2}{3}\tan^{-1}\frac{(t-1/t)}{\sqrt{3}} +C$

$I= \frac{2}{\sqrt3}\tan^{-1}\frac{(x)}{\sqrt{3(x+1)}} +C$

Option 1)

Incorrect

Option 2)

Incorrect

Option 3)

Correct

Option 4)

Incorrect

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