Trigonometric Integrals - Practice Questions & MCQ

(a) Integral of the form

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

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 $\int \frac{f(t) d t}{A t^2+B t+C}$, 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. $\int \frac{1}{a \sin x+b \cos x} d x$
2. $\int \frac{1}{a+b \sin x} d x$
3. $\int \frac{1}{a+b \cos x} d x$,
4. $\int \frac{1}{a \sin x+b \cos x+c} d x$

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
$\int \frac{1}{a^2+b t+c} d t$ which can be solved by the method we studied in previous concepts.

(c) Integrals of the form

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

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 $\boldsymbol{\lambda}, \boldsymbol{\mu}$ 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{aligned}
\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} d x+\mu \int \frac{-a \sin x+b \cos x}{a \cos x+b \sin x+c} d x \\
& \quad+\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} d x
\end{aligned}
$

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