Question

Complete the square and give a substitution (not necessarily trigonometric) which could be used to compute the integral. ∫1x2+2x+2dx

Answer 1

Answer:

Substitution of x+1

Step-by-step explanation:

We are given that

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

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

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

By using identity

$(a+b)^2=a^2+b^2+2ab$

Substitute x+1=t

Differentiate w.r.t x

dx=dt

Substitute the values

$\int\frac{1}{t^2+1^2}dx$

$\frac{1}{1}tan^{-1}\frac{t}{1}+C$

By using formula :$\int\frac{1}{x^2+a^2}dx=\frac{1}{a}tan^{-1}\frac{x}{a}+C$

$tan^{-1}(x+1)+C$