Evaluate: (x - a) (x + a) dx. - Vidyadip

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}$
$= \cos 2\text{a} \int \text{dt} – \int \sin 2\text{a}. \cot \text{t dt} = \cos 2\text{a.t} – \sin 2\text{a}. \log|\sin \text{t}|+ \text{C}$
$= \cos 2\text{a}.\text{(x + a)} – \sin 2\text{a}. \log|\sin \text{(x + a)|+ C}$
$= \text{x} \cos 2\text{a + a} \cos 2\text{a} – (\sin 2\text{a)} \log|\sin \text{(x + a)|+ C}.$

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 INTEGRALS→INTEGRALS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\limits^{\text{b}}_{\text{a}}\frac{\text{x}^{\frac{1}{\text{n}}}}{\text{x}^\frac{1}{\text{n}}+\big(\text{a}+\text{b}-\text{x}\big)^{\frac{1}{\text{n}}}}\text{ dx},\text{ n}\in\text{N},\text{n}\leq2$

Find the area under the given curves and given lines:

$y = x^2, x = 1, x = 2$ and $x-$axis
$y = x^4, x = 1, x = 5$ and $x-$axis.

A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is at most $24$. It takes $1$ hour to make a ring and $30$ minutes to make a chain. The maximum number of hours available per day is $16$. If the profit on a ring is $Rs. 300$ and that on a chain is $Rs. 190,$ find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an $\text{LPP}$ and solve it graphically.

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}}\frac{\cos^{2}\text{x}}{1+\text{e}^{\text{x}}}\text{ dx}$

Prove that the area the first quadeant enclosed by the $x-$axis, the line $\text{x}=\sqrt{3}\text{y}$ and the circle is $\frac{\pi}{3}$ .

Evaluate the following integrals:
$\int\limits^{\pi}_0\text{x}\sin^3\text{x}\text{ dx}$

Solve the following initial value problems:
$\frac{\text{dy}}{\text{dx}}+\text{y}\cot\text{x}=4\text{x }\text{cosec x},\text{ y}\Big(\frac{\pi}{2}\Big)=0$

Solve the following differential equation:
$(\text{x}+\tan\text{y})\text{dy}=\sin2\text{y dx}$

Sketch the region bounded by the curves $y = x^2 + 2, y = x, x = 0$ and $x = 1.$ Also find the area of this region.

Differentiate the following functions with respect to x:
$\sin^{-1}\big(2\text{x}^2-1\big),0<\text{x}<1$