Question

(c) Find the value of \( \int \sqrt{5 + 2x + x^2} \, dx \).

Hint:

Complete the square before integrating expressions under a square root.

Answer 1

Completethesquareintheexpression\(5+2x+x^2\): \[ 5+2x+x^2=(x+1)^2+4. \] Substitute\(u=x+1\),then\(du=dx\): \[ \int\sqrt{5+2x+x^2}\,dx=\int\sqrt{u^2+4}\,du. \] Solveusingtrigonometricsubstitution: \[ u=2\tan\theta\quad\Rightarrow\quad\sqrt{u^2+4}=2\sec\theta. \] Evaluateandsimplifytheintegral.