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**F. INTEGRATION BY REDUCTION FORMULAE**

**Ex.47 **

**Sol.**

**Ex.48 Integration of 1/(x ^{2} + k)^{n}.**

**Sol.**

Above is the reduction formula for [1/(x2 + k)^{n}]dx . By repeated application of this formula the integral shall reduce to that of

**Ex.49 Integrate (x + 2) / (2x ^{2} + 4x + 3)^{2}.**

**Sol.** Here (d/dx) (2x^{2} + 4x + 3) = 4x + 4.

Now put x + 1 = t and then applying the reduction formula, we get

**Ex.50 Integrate (2x + 3)/(x ^{2} + 2x + 3)^{2}.**

**Sol.** Here (d/dx) (x2 + 2x + 3) = 2x + 2

**Ex.51 If Im= **

**Sol.**

**Ex.52 If I _{m,n} = **

**Sol.** We have, I_{m,n} = cos^{m} x.cosnx dx

As we have cos (n – 1) x = cos nx cos x + sin nx . sinx

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