Integrate the function in Exercise: (x-a) - Vidyadip

Question

Integrate the function in Exercise:
$\frac{\sin\text{x}}{\sin(\text{x}-\text{a)}}$

✓

Answer

$\frac{\sin\text{x}}{\sin(\text{x}-\text{a)}}$
$\text{Let}\ \text{x}-\text{a}=\text{t}\Rightarrow\text{dx}=\text{dt}$
$\int\frac{\sin\text{x}}{\sin(\text{x}-\text{a)}}\text{dx}=\int\frac{\sin(\text{t}+\text{a})}{\sin\text{t}}\text{dt}$
$=\int\frac{\sin\text{t}\cos\text{a}+\cos\text{t}\sin\text{a}}{\sin\text{t}}\text{dt}$
$=\int(\cos\text{a}+\cot\text{t}\sin\text{a)}\text{dt}$
$=\text{t}\cos\text{a}+\sin\text{a}\log|\sin\text{t|}+\text{C}_{1}$
$=(\text{x}-\text{a)}\cos\text{a}+\sin\text{a}\log|\sin(\text{x}-\text{a)}|+\text{C}_{1}$
$=\text{x}\cos\text{a}+\sin\text{a}\log|\sin(\text{x}-\text{a)}|-\text{a}\cos\text{a}+\text{C}_{1}$
$=\sin\text{a}\log|\sin(\text{x}-\text{a)}|+\text{x}\cos\text{a}+\text{C}$

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 "3 Marks Question" 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

Let $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}$ and $\vec{\text{c}}=\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}}.$ Then, If $c_{1 }= 1$ and $c_{2 }= 2,$ find $c_3$ which makes $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ coplanar.

Find $\frac{d y}{d x},$ if $y = \sin^{–1} x + \sin^{–1} \sqrt{1-x^{2}}, 0 < x < 1$

Differential equation $\frac{\text{dy}}{\text{dx}}+\text{y}=0,\text{y}(0)=3$ Function $\text{y}=\text{e}^\text{-x}+2$

Check the commutativity and associativity of the following binary operations: $ ' \ ^*\ '$ on $Q$ defined by $a ^* b = (a - b)^2$ for all $a, b \in Q.$

If $\text{x}=\text{a}(\cos2\text{t}+2\text{t}\sin2\text{t})\ \text{and}\ \text{y}=\text{a}(\sin2\text{t}-2\text{t}\cos2\text{t}),$ then find $\frac{\text{d}^2\text{y}}{\text{dx}^2}.$

show that $\text{y}=\text{be}^\text{x}+\text{ce}^{2\text{x}}$ is a solution of the differential equation, $\frac{\text{d}^2\text{y}}{\text{dx}^2}-3\frac{\text{dy}}{\text{dx}}+2\text{y}=0$

Evaluate the following integrals:$\int\frac{1}{\sqrt{16-6\text{x}-\text{x}^2}}\text{ dx}$

If $\text{f(x)}=\frac{2\text{x}+3\sin\text{x}}{3\text{x}+2\sin\text{x}},\text{ x}\neq0$ is continuous at x = 0, then find f(0).

Check the commutativity and associativity of the following binary operations:
'o' on Q defined by $\text{a o b}=\frac{\text{ab}}{2}$ for all a, b ∈ Q.

Evaluate the following integrals:
$\int\frac{\text{x}}{(\text{x}^2+4)\sqrt{\text{x}^2+9}}\text{ dx}$