Evaluate the following integrals: 1 (3+2 ) dx - Vidyadip

Question

Evaluate the following integrals: $\int\frac{1}{\sin\text{x}(3+2\cos\text{x})}\ \text{dx}$

✓

Answer

Let $\text{I}=\int\frac{1}{\sin\text{x}(3+2\cos\text{x})}\ \text{dx}$
$=\frac{\sin\text{x dx}}{\sin^2\text{x}(3+2\cos\text{x })}$
$=\frac{\sin\text{x dx}}{(1-\cos^2\text{x})(3+2\text{x})}$
Let $\cos\text{x}=\text{t}$
$\Rightarrow-\sin\text{x dx}=\text{dt}$
$\therefore\text{I}=\int\frac{\text{dt}}{(\text{t}^2-1)(3+2\text{t})}$
Now,
Let $\frac{1}{(\text{t}^2-1)(3+2\text{t})}=\frac{\text{A}}{\text{t}-1}+\frac{\text{B}}{\text{t}+1}+\frac{\text{C}}{3+2\text{t}}$
$\Rightarrow 1 = A(t + 1)(3 + 2t) + B(t - 1)(3 + 2t) + C (t^2- 1)$
Put $t = -1$
$\Rightarrow 1 = -2B$
$\Rightarrow\text{A}=\frac{1}{10}$
Put $\text{t}=-\frac{3}{2}$
$\Rightarrow1=\frac{5}{4}\text{C}$
$\Rightarrow\text{C}=\frac{4}{5}$
Thus,
$\text{I}=\frac{1}{10}\int\frac{\text{dt}}{\text{t}-1}-\frac{1}{2}\int\frac{\text{dt}}{\text{t}+1}+\frac{5}{4}\int\frac{\text{dt}}{3+2\text{t}}$
$=\frac{1}{10}\log|\text{t}-1|=\frac{1}{2}\log|\text{t}+1|+\frac{2}{5}\log|3+2\text{t}|+\text{C}$
Hence,
$\text{I}=\frac{1}{10}\log|\cos\text{x}-1|-\frac{1}{2}\log|\cos\text{x}+1|+\frac{2}{5}\log|3+2\cos\text{x}|+\text{C}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Indefinite Integrals→Indefinite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

$\text{If}\cos\text{y}=\text{x}\cos(\text{a}+\text{y}),\ \text{with}\cos\text{a}\neq\pm1,\ \text{prove that}\frac{\text{dy}}{\text{dx}}=\frac{\cos^2(\text{a}+\text{y})}{\sin\text{a}}$

There are three coins. One is two headed coin, another is a biased coin that comes up heads $75\%$ of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?

Find the equation of the plane passing through the intersection of the planes x - 2y + z = 1 and 2x + y + z= 8 and parallel to the line with direction ratios proportional to 1, 2, 1. Also, find the perpendicular distance of (1, 1, 1) from this plane.

The probability that a certain person will buy a shirt is 0.2, the probability that he will buy a trouser is 0.3, and the probability that he will buy a shirt given that he buys a trouser is 0.4. Find the probability that he will buy both a shirt and a trouser. Find also the probability that he will buy a trouser given that he buys a shirt.

Examine the differentiability of f, where f is defined by:
$\text{f(x)}=\begin{cases}1+\text{x},&\text{if x}\leq2\\5-\text{x},&\text{if x}>2\end{cases}$
at x = 2.

Prove that the curves $y^2 = 4x$ and $x^2 + y^2 - 6x + 1 = 0$ touch each other at the point $(1, 2).$

Solve the following differential equation
$\cos\text{x }\frac{\text{dy}}{\text{dx}}-\cos2\text{x}=\cos3\text{x}$

Solve the following equation for x:
$\tan^{-1}2\text{x}+\tan^{-1}3\text{x}=\text{n}\pi+\frac{3\pi}{4}$

Integrate the function in Exercise:
$\frac{1}{\text{x}^{\frac{1}{2}}+\text{x}^{\frac{1}{3}}}$
$\Bigg[\text{Hint:}\frac{1}{\text{x}^{\frac{1}{2}}+\text{x}^{\frac{1}{3}}}=\frac{1}{\text{x}^{\frac{1}{3}}\bigg(1+\text{x}^{\frac{1}{6}}\bigg)},\text{put}\ \text{x}=\text{t}^{6}\Bigg]$

If $f(x) = x^3 + 7x^2 + 8x - 9,$ find $f(4).$