Evalute the following integrals: 1 (x+a) (x+b) dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{1}{\cos(\text{x}+\text{a})\cos(\text{x}+\text{b})}\text{dx}$

✓

Answer

$\int\frac{1}{\cos(\text{x}+\text{a})\cos(\text{x}+\text{b})}\text{dx}$
Multiplying and Dividing by $\sin\big[(\text{x}+\text{b})-(\text{x}+\text{a})\big],$ we get
$=\int\frac{1}{\sin\big[(\text{x}+\text{b})-(\text{x}+\text{a})\big]}\times\frac{\sin\big[(\text{x}+\text{b})-(\text{x}+\text{a})\big]}{\cos(\text{x}+\text{a})\cos(\text{x}+\text{b})}\text{dx}$
$=\int\frac{1}{\sin(\text{b}-\text{a})}\times\frac{\sin\big[(\text{x}+\text{b})-(\text{x}+\text{a})\big]}{\cos(\text{x}+\text{a})\cos(\text{x}+\text{b})}\text{dx}$
$=\frac{1}{\sin(\text{b}-\text{a})}\int\frac{\sin(\text{x}+\text{b})\cos(\text{x}+\text{a})-\sin(\text{x}+\text{a})\cos(\text{x}+\text{b})}{\cos(\text{x}+\text{a})\cos(\text{x}+\text{b})}\text{dx}$
$=\frac{1}{\sin(\text{b}-\text{a})}\Big[\int\frac{\sin(\text{x}+\text{b})}{\cos(\text{x}+\text{b})}\text{dx}-\int\frac{\sin(\text{x}+\text{a})}{\cos(\text{x}+\text{a})}\text{dx}\Big]$
$=\frac{1}{\sin(\text{b}-\text{a})}\Big[\int\tan(\text{x}+\text{b})\text{dx}-\int\tan(\text{x}+\text{a})\text{dx}\Big]$
$=\frac{1}{\sin(\text{b}-\text{a})}\big[\log(\sec(\text{x}+\text{b}))-\log(\sec(\text{x}+\text{a}))\big]+\text{C}$
$=\frac{1}{\sin(\text{b}-\text{a})}\bigg[\log\Big(\frac{\sec(\text{x}+\text{b})}{\sec(\text{x}+\text{a})}\Big)\bigg]+\text{C}$
Hence, $\int\frac{1}{\cos(\text{x}+\text{a})\cos(\text{x}+\text{b})}\text{dx}=\frac{1}{\sin(\text{b}-\text{a})}\bigg[\log\Big(\frac{\sec(\text{x}+\text{b})}{\sec(\text{x}+\text{a})}\Big)\bigg]+\text{C}$

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 "4 Marks" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the vector equation of the plane passing through three point with position vectors $\hat{\text{i}}+\hat{\text{j}}-2\hat{\text{k}},2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and $\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}.$ Also, find coordinates of the point of intersection of this plane and the line $\vec{\text{r}}=3\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}+\lambda(2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}).$

By using the properties of definite integral, evaluate the integral in Exercise:
$\int^{\frac{\pi}{4}}_{0}\log(1+\tan\text{x})\text{dx}$

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

Find the angle between the following pairs of lines:

$\frac{\text{x}-1}{2}=\frac{\text{y}-2}{3}=\frac{\text{z}-3}{-3}$ and $\frac{\text{x}+3}{-1}=\frac{\text{y}-5}{8}=\frac{\text{z}-1}{4}$

Show that area of the parallelogram whose diagonals ate given by $\vec{\text{a}}$ and $\vec{\text{b}}$ is $\frac{|\vec{\text{a}}\times\vec{\text{b}}|}{2}.$ Also, find the area of the parallelogram, whose diagonals are $2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and $\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}.$

In interval $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$, Find the difference in maximum value and minimum value of function $f(x)=\sin 2 x-x$.

Prove the following results:
$2\tan^{-1}\frac{1}{5}+\tan^{-1}\frac{1}{8}=\tan^{-1}\frac{4}{7}$

A function f : R → R satisfies the equation f(x + y) = f(x)f(y) fot all x, y ∈ R, f(x) ≠ 0. Suppose that the function is differentiable at x = 0 and f'(0) = 2. Prove that f'(x) = 2 f(x).

Evaluvate the following intregals:
$\int\frac{1}{\text{p}+\text{q}\tan\text{x}}\ \text{dx}$

A furniture manufacturing company plans to make two products : chairs and tables. From its available resources which consists of 400 square feet to teak wood and 450 man hours. It is known that to make a chair requires 5 square feet of wood and 10 man-hours and yields a profit of Rs. 45, while each table uses 20 square feet of wood and 25 man-hours and yields a profit of Rs. 80. How many items of each product should be produced by the company so that the profit is maximum?