# Part - 12 Lecture - 4 Chapter 7 Integrals

“Whether you think you can or whether you think you can’t, you’re right!”
— Henry Ford —

INTEGRATION OF ALGEBRAIC FUNCTIONS

THIRD METHOD

We apply the third method only if the first two methods are failed.
$px+q=A(ax^2+bx+c)^\prime+B$
It can be applied if the question is in $\int\frac{px+q}{ax^2+bx+c}dx$,$\int\frac{px+q}{\sqrt{ax^2+bx+c}}dx$,$\int{(px+q)\sqrt{ax^2+bx+c}dx}$.

24. $\int (3x-2)\sqrt{x^2+x+1}dx=(x^2+x+1)^\frac{3}{2}-\frac{7}{2}\left[\left(\frac{2x+1}{4}\right)\sqrt{x^2+x+1}+\frac{3}{8}log{\left|x+\frac{1}{2}+\sqrt{x^2+x+1}\right|}\right]+C$

25. $\int(x-5)\sqrt{x^2+x}dx=\frac{1}{3}(x^2+x)^\frac{3}{2}-\frac{11}{2}\left\{\frac{2x+1}{4}\sqrt{x^2+x}-\frac{1}{8}log{\left|\left(x+\frac{1}{2}\right)+\sqrt{x^2+x}\right|}\right\}+C$