## Part - 10 Lecture - 4 Chapter 7 Integrals

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INTEGRATION OF ALGEBRAIC FUNCTIONS

SECOND METHOD
In the second method, we use integrals of some particular functions to find the integrals of our rational functions.

We use Perfect Square in cases like $$\int\frac{dx}{ax^2+bx+c}$$, $$\int\frac{dx}{\sqrt{ax^2+bx+c}}$$, $$\int{\sqrt{ax^2+bx+c}dx}$$

15. $$\int\frac{dx}{9x^2+6x+5}=\frac{1}{6}{tan}^{-1}{\left(\frac{3x+1}{2}\right)}+C$$

16. $$\int\frac{dx}{\sqrt{5x^2-2x}}=\frac{1}{\sqrt5}log{\left|x-\frac{1}{5}+\sqrt{x^2-\frac{2x}{5}}\right|}+C$$

17. $$\int\frac{dx}{\sqrt{9x-4x^2}}=\frac{1}{2}{sin}^{-1}{\left(\frac{8x-9}{9}\right)}+C$$

18. $$\int{\sqrt{1-4x-x^2}dx}=\frac{5}{2}{sin}^{-1}{\left(\frac{x+2}{\sqrt5}\right)}+\frac{x+2}{2}\sqrt{1-4x-x^2}+C$$

19. $$\int {\frac{1}{\sqrt{(x-a)(x-b)}}}dx=\log{\left|x-\frac{a-b}{2}+\sqrt{(x-a)(x-b)}\right|}+C$$