JEE Main & Advanced Mathematics Indefinite Integrals Integration of Trigonometric Functions

(1) Integral of the form \[\int{si{{n}^{m}}x\,co{{s}^{n}}x\,dx}\]: (i) To evaluate the integrals of the form \[I=\int{{{\sin }^{m}}x\,{{\cos }^{n}}x\,dx,}\] where \[m\] and \[n\] are rational numbers.

(a) Substitute \[\sin x=t,\] if \[n\] is odd;

(b) Substitute cos x = t, if m is odd;

(c) Substitute \[\tan x=t,\] if \[m+n\] is a negative even integer; and

(d) Substitute \[\cot x=t,\] if \[\frac{1}{2}(n-1)\] is an integer.

(e) If \[m\] and \[n\] are rational numbers and \[\left( \frac{m+n-2}{2} \right)\] is a negative integer, then substitution \[\cos x=t\] or \[\tan x=t\] is found suitable.

(ii) Integrals of the form \[\int{R(\sin x,\,\cos x)\,dx,}\] where \[R\] is a rational function of \[\sin x\] and \[\cos x,\] are transformed into integrals of a rational function by the substitution \[\tan \frac{x}{2}=t,\]where \[-\pi <x<\pi .\]  This is the so called universal substitution. Sometimes it is more convenient to make the substitution \[\cot \frac{x}{2}=t\] for \[0<x<2\pi .\]

The above substitution enables us to integrate any function of the form \[R(\sin x,\,\cos x).\] However, in practice, it sometimes leads to extremely complex rational function. In some cases, the integral can be simplified by :

(a) Substituting \[\sin x=t,\] if the integral is of the form \[\int{R(\sin x)\cos x\,dx}\].

(b) Substituting \[+N(c\cos x-d\sin x)+P.\] if the integral is of the form \[\int{R(\cos x)\sin x\,dx}\].

(c) Substituting tan \[x=t\], i.e., \[dx=\frac{dt}{1+{{t}^{2}}},\] if the integral is dependent only on \[\tan x.\]

(d) Substituting \[\cos x=t\], if \[R(-\sin x,\,\cos x)=-R(\sin x,\,\cos x)\]

(e) Substituting \[\sin x=t\], if \[R(\sin x,-\,\cos x)=-R(\sin x,\,\cos x)\]

(f) Substituting \[\tan x=t\], if \[R(-\sin x,-\,\cos x)=-R(\sin x,\,\cos x)\]

(2) Reduction formulae for special cases

(i) \[\int{{{\sin }^{n}}x\,dx=\frac{-\cos x\,.\,{{\sin }^{n-1}}x}{n}}+\frac{n-1}{n}\int{{{\sin }^{n-2}}x\,dx}\]       

(ii) \[\int{{{\cos }^{n}}x\,dx=\frac{\sin x{{\cos }^{n-1}}x}{n}}+\frac{n-1}{n}\int{{{\cos }^{n-2}}x\,dx}\]

(iii) \[\int{{{\tan }^{n}}x\,dx=\frac{{{\tan }^{n-1}}x}{n-1}-\int{{{\tan }^{n-2}}}x\,dx}\]

(iv) \[\int{{{\cot }^{n}}x\,dx=\frac{-1}{n-1}{{\cot }^{n-1}}x-\int{{{\cot }^{n-2}}x\,dx}}\]

(v) \[\int{{{\sec }^{n}}x\,dx=\frac{1}{(n-1)}\left[ {{\sec }^{n-2}}x.\,\tan x+(n-2)\int{{{\sec }^{n-2}}}x\,dx \right]}\]

(vi) \[\int{\text{cose}{{\text{c}}^{n}}}xdx=\frac{1}{(n-1)}[-\text{cose}{{\text{c}}^{n-2}}x\cot x+(n-2)\int{\text{cose}{{\text{c}}^{n-2}}xdx}]\]

(vii) \[\int{{{\sin }^{p}}x{{\cos }^{q}}x\,dx=-\frac{{{\sin }^{q+1}}x.\,{{\cos }^{p-1}}x}{p+q}}\]\[\sin x=\frac{2\tan (x/2)}{1+{{\tan }^{2}}(x/2)}\]

(viii) \[\int{{{\sin }^{p}}x{{\cos }^{q}}x\,dx=\frac{{{\sin }^{p+1}}x\,.{{\cos }^{q-1}}x}{p+q}}\] \[+\frac{p-1}{p+q}\int{{{\sin }^{p}}x.{{\cos }^{q-2}}x\,dx}\]

(ix) \[\int{\frac{dx}{{{({{x}^{2}}+k)}^{n}}}=\frac{x}{k(2n-2)\,{{({{x}^{2}}+k)}^{n-1}}}+\frac{(2n-3)}{k(2n-2)}}\,\int{\frac{dx}{{{({{x}^{2}}+k)}^{n-1}}}}\]