Trigonometric Integrals - Practice Questions & MCQ

(a) Integral of the form

1. $\int \frac{1}{a{\cos }^{2}x+b{\sin }^{2}x}dx$
2. $\int \frac{1}{a+b{\sin }^{2}x}dx$
3. $\int \frac{1}{a+b{\cos }^{2}x}dx$
4. $\int \frac{1}{a+b{\sin }^{2}x+c{\cos }^{2}x}dx$

Working Rule:

Step 1: Divide the numerator and denominator both by ${\cos }^{2}x$.
Step 2 : Put $\tan \mathrm{x}=\mathrm{t},{\sec }^2\mathrm{xdx}=\mathrm{dt}$
This substitution will convert the trigonometric integral into an algebraic integral.

After employing these steps the integral will reduce to the form $\int \frac{f(t)dt}{A{t}^2+Bt+C}$, where f(t) is a polynomial in t.

This integral can be evaluated by methods we studied in previous concepts.

(b) Integral of the form

1. $\int \frac{1}{a\sin x+b\cos x}dx$
2. $\int \frac{1}{a+b\sin x}dx$
3. $\int \frac{1}{a+b\cos x}dx$,
4. $\int \frac{1}{a\sin x+b\cos x+c}dx$

Working Rule:

Write sin x and cos x in terms of tan (x/2) and then substitute for tan (x/2) = t

i.e.

$\sin x=\frac{2\tan x/2}{1+{\tan }^{2}x/2}\text{ and }\cos x=\frac{1-{\tan }^{2}x/2}{1+{\tan }^{2}x/2}$
replace, $\tan (\mathrm{x}/2)$ with $t$
by performing these steps the integral reduces to the form
$\int \frac{1}{a^2+bt+c}dt$ which can be solved by the method we studied in previous concepts.

(c) Integrals of the form

1. $\int \frac{p\cos x+q\sin x+r}{a\cos x+b\sin x+c}dx$
2. $\int \frac{p\cos x+q\sin x}{a\cos x+b\sin x}dx$

Working Rule:

Express numerator as $\lambda$ (denominator $)+\mu ($ differentiation of denominator $)+\gamma$
$\Rightarrow (\mathrm{p}\cos \mathrm{x}+\mathrm{q}\sin \mathrm{x}+\mathrm{r})=\lambda (\mathrm{a}\cos \mathrm{x}+\mathrm{b}\sin \mathrm{x}+\mathrm{c})+\mu (-\mathrm{a}\sin \mathrm{x}+\mathrm{b}\cos \mathrm{x})+\gamma$
where $\mathit{\lambda },\mathit{\mu }$ and y are constants to be determined by comparing the coefficients of $\sin \mathrm{x},\cos \mathrm{x}$, and constant terms on both sides.
$\begin{array}{rl}\int \frac{\mathrm{p}\cos \mathrm{x}+\mathrm{q}\sin \mathrm{x}+\mathrm{r}}{\mathrm{a}\cos \mathrm{x}+\mathrm{b}\sin \mathrm{x}+\mathrm{c}}\mathrm{dx}=& \int \frac{\lambda (\mathrm{a}\cos \mathrm{x}+\mathrm{b}\sin \mathrm{x}+\mathrm{c})+\mu (-\mathrm{a}\sin \mathrm{x}+\mathrm{b}\cos \mathrm{x})+\gamma }{\mathrm{a}\cos \mathrm{x}+\mathrm{b}\sin \mathrm{x}+\mathrm{c}}\mathrm{dx}\\ =& \lambda \int \frac{a\cos x+b\sin x+c}{a\cos x+b\sin x+c}dx+\mu \int \frac{-a\sin x+b\cos x}{a\cos x+b\sin x+c}dx\\ & +\int \frac{\mu }{\mathrm{a}\cos \mathrm{x}+\mathrm{b}\sin \mathrm{x}+\mathrm{c}}\mathrm{dx}\\ =& \lambda x+\mu \ln |a\cos x+b\sin x+c|+\int \frac{\mu }{a\cos x+b\sin x+c}dx\end{array}$

The last integral on RHS can be evaluated by the methods we studied in the previous concept.