Integration of Trigonometric Function

Here you will learn integration of trigonometric function with examples.

Let’s begin –

Integration of Trigonometric Function

(i)  \(\int\) \(sin^mxcos^nx\)

Case-1 : When both m & n \(\in\) 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 \(\int\) \(sin^3xcos^5x\) dx

Solution : We have \(\int\) \(sin^3xcos^5x\) dx,

Put cosx = t; -sinx dx = dt

so that I = – \(\int\) \((1 – t^2).t^5\) dt

  = \(\int\) \((t^7 – t^5)\) dt = \(t^8\over 8\) – \(t^6\over 6\) = \(cos^8x\over 8\) – \(cos^6x\over 6\) + C

(ii)  \(\int\) \(dx\over {a + bsin^2x}\) OR \(\int\) \(dx\over {a + bcos^2x}\) OR \(\int\) \(dx\over {asin^2x + bsinxcosx + ccos^2x}\)

Divide Numerator and Denominator by \(cos^2x\) & put tanx = t.

Example : Evaluate : \(\int\) \(dx\over {(2sinx + 3cosx)}^2\)

Solution : Divide numerator and denominator by \(cos^2x\)

\(\therefore\)   I = \(\int\) \(sec^2x dx\over {(2sinx + 3cosx)}^2\)

Let 2tanx + 3 = t,       \(\therefore\) 2\(sec^2x\) dx = dt

I = \(1\over 2\) \(\int\) \(dt\over {t^2}\) = -\(1\over 2t\) + C = -\(1\over {2(2tanx+3)}\) + C

(iii)  \(\int\) \(dx\over {a + bsinx}\) OR \(\int\) \(dx\over {a + bcosx}\) OR \(\int\) \(dx\over {a + bsinx + ccosx}\)

convert sines and cosines into their respective tangents of half the angles and put tan\(x\over 2\) = t

In this case sinx = \(2t\over {1+t^2}\), cosx = \({1-t^2}\over {1+t^2}\), x = 2\(tan^{-1}t\); dx = \(2dt\over {1+t^2}\)

(iv)  \(\int\) \(acosx + bsinx + c\over {pcosx + qsinx + r}\)

Express Numerator = a(Denominator) + b\(d\over dx\)(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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