Evalute the following integrals: (x-a) (x-b) dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{\sin(\text{x}-\text{a})}{\sin(\text{x}-\text{b})}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{\sin(\text{x}-\text{a})}{\sin(\text{x}-\text{b})}\text{dx}$
Putting x - b = t
⇒ x = b + t
& dx = dt
$\therefore\text{I}=\int\frac{\sin(\text{b}+\text{t}-\text{a})}{\sin\text{t}}\text{dt}$
$=\int\frac{\sin\big\{(\text{b}-\text{a})+\text{t}\big\}}{\sin\text{t}}\text{dt}$
$=\int\frac{\sin(\text{b}-\text{a})\cos\text{t}}{\sin\text{t}}+\int\frac{\cos(\text{b}-\text{a})\sin\text{t}}{\sin\text{t}}\text{dt}$
$=\int\sin(\text{b}-\text{a})\cot\text{t dt}+\int\cos(\text{b}-\text{a})\text{dt}$
$=\sin(\text{b}-\text{a})\text{in}|\sin\text{t}\big|+\text{t}\cos(\text{b}-\text{a})+\text{C}$
$=\sin(\text{b}-\text{a})\text{in}|\sin(\text{x}-\text{b})|+(\text{x}-\text{b})\cos(\text{b}-\text{a})+\text{C}$
$\big[\because\text{t}=\text{x}-\text{b}\big]$

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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Solve the following differential equations:
$\text{y}\sqrt{1+\text{x}^2}+\text{x}\sqrt{1+\text{y}^2}\ \frac{\text{dy}}{\text{dx}}=0$

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\sin\text{x}\sin2\text{x}\sin3\text{x}\sin4\text{x}$

A pair of dice is thrown. Find the probability of getting the sum 8 or more, if 4 appears on the first die.

Ishan wants to donate a rectangular plot of land for a school in his village. When he was asked to give dimensions of the plot, he told that if its length is decreased by 50 m and breadth is increased by 50 m, then its area will remain same, but if length is decreased by 10 m and breadth is decreased by 20 m, then its area will decrease by $5300 \text{ m}^{2}.$ Using matrices, find the dimensions of the plot. Also give reason why he wants to donate the plot for a school.

Evaluvate the following intregals
$\int\frac{2\text{x}+3}{\sqrt{\text{x}^2+4\text{x}+5}}\text{dx}$

The function $\text{f(x)}=\begin{cases}\frac{\text{x}^2}{\text{a}},&\text{if }0\leq\text{ x}<1\\\text{a},&\text{if }1\leq\text{x}<\sqrt{2}\\\frac{2\text{b}^2-4\text{b}}{\text{x}^2},&\text{if }\sqrt{2}\leq\text{x}<\infty\end{cases}$ is continuous on $(0,\infty),$ then find the most suitable value of a and b.

Given $\vec{\text{a}}=\frac{1}{7}\big(2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}\big),\vec{\text{b}}=\frac{1}{7}\big(3\hat{\text{i}}-6\hat{\text{j}}+2\hat{\text{k}}\big),$$\vec{\text{c}}=\frac{1}{7}\big(6\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\big),\hat{\text{i}},\hat{\text{j}},\hat{\text{k}}$
being a right handed orthogonal system of unit vector in spece, show that $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ is also another system.

A letter is known to have come either from LONDON or CLIFTON. On the envelope just two consecutive letters ON are visible. What is the probability that the letter has come from,
LONDON.

Draw a rough sketch of the region {$(x, y) : y^2 < 3x, 3x^2 + < 16$} and find the area by the region using mwthod of integration.

Show that the normal vector to the plane 2x + 2y + 2z = 3 is equally inclined to the coordinate axes.