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Integration of Trigonometric Function

Here you will learn integration of trigonometric function with examples.

Let’s begin –

Integration of Trigonometric Function

(i)  sinmxcosnx

Case-1 : When both m & n natural numbers.

(a)  If one of them is odd, then substitute for the term of even power.

(b)  If both are odd, substitute either of them.

(c)  If both are even, use trigonometric identities to convert integrand into cosines of multiple angles.

Case-2 : When m + n is a negative even integer.

In this case the best substitution is tanx = t.

Example : Evaluate sin3xcos5x dx

Solution : We have sin3xcos5x dx,

Put cosx = t; -sinx dx = dt

so that I = – (1t2).t5 dt

  = (t7t5) dt = t88t66 = cos8x8cos6x6 + C

(ii)  dxa+bsin2x OR dxa+bcos2x OR dxasin2x+bsinxcosx+ccos2x

Divide Numerator and Denominator by cos2x & put tanx = t.

Example : Evaluate : dx(2sinx+3cosx)2

Solution : Divide numerator and denominator by cos2x

  I = sec2xdx(2sinx+3cosx)2

Let 2tanx + 3 = t,       2sec2x dx = dt

I = 12 dtt2 = -12t + C = -12(2tanx+3) + C

(iii)  dxa+bsinx OR dxa+bcosx OR dxa+bsinx+ccosx

convert sines and cosines into their respective tangents of half the angles and put tanx2 = t

In this case sinx = 2t1+t2, cosx = 1t21+t2, x = 2tan1t; dx = 2dt1+t2

(iv)  acosx+bsinx+cpcosx+qsinx+r

Express Numerator = a(Denominator) + bddx(Denominator) + c & proceed.

Hope you learnt integration of trigonometric function, learn more concepts of integration and practice more questions to get ahead in competition. Good Luck!

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