Question
Evaluate: $\int\frac{\sin(\text{x} - \text{a})}{\sin(\text{x + a})}\text{ dx}.$
✓
Answer
Let $\text{I} =\int\frac{\sin(\text{x - a )}}{\sin\text{(x +a )}}\text{dx}$
Let x + a =t $\Rightarrow$x =t – a
$\Rightarrow$dx = dt
$\therefore\text{I} = \int\frac{\sin(\text{t} - 2\text{a})}{\sin\text{t}}\text{dt}$
$ =\int\frac{\sin\text{t}.\cos2\text{a} - \cos\text{t}.\sin2\text{a}}{\sin\text{t}}\text{dt}$
$= \cos 2\text{a} \int \text{dt} – \int \sin 2\text{a}. \cot \text{t dt} = \cos 2\text{a.t} – \sin 2\text{a}. \log|\sin \text{t}|+ \text{C}$
$= \cos 2\text{a}.\text{(x + a)} – \sin 2\text{a}. \log|\sin \text{(x + a)|+ C}$
$= \text{x} \cos 2\text{a + a} \cos 2\text{a} – (\sin 2\text{a)} \log|\sin \text{(x + a)|+ C}.$
Let x + a =t $\Rightarrow$x =t – a
$\Rightarrow$dx = dt
$\therefore\text{I} = \int\frac{\sin(\text{t} - 2\text{a})}{\sin\text{t}}\text{dt}$
$ =\int\frac{\sin\text{t}.\cos2\text{a} - \cos\text{t}.\sin2\text{a}}{\sin\text{t}}\text{dt}$
$= \cos 2\text{a} \int \text{dt} – \int \sin 2\text{a}. \cot \text{t dt} = \cos 2\text{a.t} – \sin 2\text{a}. \log|\sin \text{t}|+ \text{C}$
$= \cos 2\text{a}.\text{(x + a)} – \sin 2\text{a}. \log|\sin \text{(x + a)|+ C}$
$= \text{x} \cos 2\text{a + a} \cos 2\text{a} – (\sin 2\text{a)} \log|\sin \text{(x + a)|+ C}.$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
There are two factories located one at place P and the other at place Q. From these locations, a certain commodity is to be delivered to each of the three depots situated at A, B and C. The weekly requirements of the depots are respectively 5, 5 and 4 units of the commodity while the production capacity of the factories at P and Q are respectively 8 and 6 units. The cost of transportation per unit is given below:
In the following, determine the values of constants involved in the definition so that the given function is continuous:
$\text{f(x)}=\begin{cases}4,&\text{if }\text{ x}\leq-1\\\text{ax}^2+\text{b},&\text{if }-1<\text{ x}<0\\\cos\text{x},&\text{if }\text{ x}\geq0\end{cases}$
→$\text{f(x)}=\begin{cases}4,&\text{if }\text{ x}\leq-1\\\text{ax}^2+\text{b},&\text{if }-1<\text{ x}<0\\\cos\text{x},&\text{if }\text{ x}\geq0\end{cases}$
$\text{Evaluate:} \int\limits_{-a}^{a} \sqrt\frac{{a - x}}{a + x} {dx}$
→
Evaluate the following integrals:
$\int\frac{\text{x}^2}{(\text{a}-\text{x}^2)^{\frac{3}{2}}}\text{ dx}$
→$\int\frac{\text{x}^2}{(\text{a}-\text{x}^2)^{\frac{3}{2}}}\text{ dx}$
If xm + yn = 1, Prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{my}}{\text{nx}}$
→Find one-parameter families of solution curves of the following differential equation: (or solve the following differential equation)
→$\frac{\text{dy}}{\text{dx}}-\text{y}=\cos2\text{x}$
Find the area of the bounded by $\text{y}=\sqrt{\text{x}}$ and y2 = x.
→A balloon in the form of a right circular cone surmounted by a hemisphere, having a diametre equal to the height of the cone, is being inflated. How fast is its volume changing with respect to its total height h, when h = 9cm.
If $\int_{0}^\limits{\text{k}}\frac{1}{2+8\text{x}^2}\text{ dx}=\frac{\pi}{16},$ find the value of k.
→Evaluate the following definite integrals:
$\int_{0}^\limits{\pi}\text{e}^{2\text{x}}\sin\Big(\frac{\pi}{4}+{\text{x}}\Big)\text{dx}$
→$\int_{0}^\limits{\pi}\text{e}^{2\text{x}}\sin\Big(\frac{\pi}{4}+{\text{x}}\Big)\text{dx}$