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

Question

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

✓

Answer

Let $\text{I}=\int\frac{\cos\text{x}}{\cos(\text{x}-\text{a})}\text{dx}$
Putting x - a = t
⇒ x = a + t
⇒ dx = dt
$\therefore\text{I}=\int\frac{\cos(\text{a}+\text{t})\text{dt}}{\cos\text{t}}$
$=\frac{\cos\text{a}\cos\text{t}}{\cos\text{t}}-\frac{\sin\text{a}\sin\text{t}}{\cos\text{t}}\text{dt}$
$=\int\big(\cos\text{a}-\sin\text{a}\tan\text{t}\big)\text{dt}$
$=\text{t}\cos\text{a}-\sin\text{a ln}|\sec\text{t}|+\text{C}$
$=(\text{x}-\text{a})\cos\text{a}-\sin\text{a ln}|\sec(\text{a}-\text{a})|+\text{C}\ \big[\text{t}=\text{x}-\text{a}\big]$

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 "3 Marks Question" 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

Define a binary operation$ *$ on the set $\{0, 1, 2, 3, 4, 5\}$ as $\text{a}*\text{b}=\begin{cases}\text{a + b},&\text{if a + b}<6\\\text{a + b}-6&\text{if a + b}\geq6\end{cases}$
Show that zero is the identity for this operation and each element a of the set is invertible with $6 – a$ being the inverse of a.

Given, $\text{A}=\begin{bmatrix}2&4&0\\3&9&6\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&4\\2&8\\1&3\end{bmatrix}$ is $(\text{AB})'=\text{B}'\text{A}'\ ?$

Two cards are drawn successively with replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of aces.

Assume that on an average one telephone number out of 15 called between 2 P.M. and 3 P.M. on week days is busy. What is the probability that if six randomly selected telephone numbers are called, at least 3 of them will be busy?

Find the points on the line $\frac{\text{x}+2}{3}=\frac{\text{y}+1}{2}=\frac{\text{z}-3}{2}$ at a distance of 5 units from the point P(1, 3, 3).

Evaluate the following integrals:
$\int^\limits1_0\frac{1-\text{x}^2}{(1+\text{x}^2)^2}\text{ dx}$

Find $\frac{\text{dy}}{\text{dx}}$ of the functions expressed in parametric:
$\sin\text{x}=\frac{2\text{t}}{1+\text{t}},\ \tan\text{y}=\frac{2\text{t}}{1-\text{t}^2}.$

Find the value of $\lambda$ so that the following vectors are coplanar:
$\vec{\text{a}}=2\hat{\text{i}}-\hat{\text{j}}++\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}},\vec{\text{c}}=\lambda\hat{\text{i}}+\lambda\hat{\text{j}}+5\hat{\text{k}}$

Find the angle between two vectors $\vec{\text{a}}$ and $\vec{\text{b},}$ if $\big|\vec{\text{a}}\times\vec{\text{b}}\big|=\vec{\text{a}}.\vec{\text{b}}.$

Solve for x and y:
$\text{x}\begin{bmatrix}2\\1\end{bmatrix}+\text{y}\begin{bmatrix}3\\5\end{bmatrix}+\begin{bmatrix}-8\\-11\end{bmatrix}=0.$