Evaluate the following integrals: .log ( + )dx - Vidyadip

Question

Evaluate the following integrals:
$\int\sec\text{x}.\text{log} (\sec\text{x}+\tan\text{x})\text{dx}$

✓

Answer

$\int\sec\text{x}.\text{log} (\sec\text{x}+\tan\text{x})\text{dx}$
$\text{Let }\text{ log}(\sec\text{x}+ \tan\text{x})=\text{t}$
$\Rightarrow \frac{(\sec\text{x} \tan\text{x}+\sec^{2}\text{x})} {(\sec\text{x} +\tan\text{x})}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow \frac{\sec\text{x} (\sec\text{x}+\tan\text{x})} {(\sec\text{x} +\tan\text{x})}\text{dx}=\text{dt}$
$\text{Now},\int\sec \text{x}.\text{log}(\sec\text{x}+\tan\text{x})\text{dx}$
$=\int \text{t}.\text{dt}$
$=\frac{\text{t}^{2}}{2}+\text{C}$
$=\frac{[\text{log}(\sec\text{x}+\tan\text{x})]^2}{2}+\text{C}$

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 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

Find the distance of the point (2, 4, -1) from the line $\frac{\text{x}+5}{1}=\frac{\text{y}+3}{4}=\frac{\text{z}-6}{-9}.$

The probability of a man hitting a target is 1/4. If he fires 7 times, what is the probability of his hitting the target at least twice?

In a 20-question true-false examination, suppose a student tosses a fair coin to determine his answer to each question. For every head, he answers 'true' and for every tail, he answers 'false'. Find the probability that he answers at least 12 questions correctly.

Evaluate the following integrals:
$\int\frac{\cos\text{x}}{1-\cos\text{x}}\text{dx}\text{ or }\int\frac{\cot\text{x}}{\text{cosec x}-\cot\text{x}}\text{dx}$

Prove that $\int_a^b f(a+b-x) d x=\int_a^b f(x) d x$ and using it find $\int_4^9 \frac{f(x)}{f(x)+f(13-x)} d x$.

Let S be the set of all rational numbers of the for $\frac{\text{m}}{\text{n}},$ where $\text{m}\in\text{Z}$ and n = 1, 2, 3. Prove that * on sdefined by a * b = ab is not a binary operation.

There are two boxes, namely $\text{Box-I}$ and $\text{Box-II}. \text{Box-I}$ contains $3$ red and $6$ black balls. $\text{Box-II}$ contains $5$ red and $5$ black balls. One of the two boxes, is selected at random and a ball is drawn at random. The ball drawn is found to be red. Find the probability that this red ball comes out from $\text{Box-II}$

Find $\frac{\text{dy}}{\text{dx}}$ of the functions given in Exercise:
$\text{xy}=\text{e}^{(\text{x}-\text{y})}$

Find the area of the parallelogram whose adjacent sides are determined by the vectors $\vec{a}=\hat{i}-\hat{j}+3 \hat{k}$ and $\vec{b}=2 \hat{i}-7 \hat{j}+\hat{k}$.

Consider the binary operations * : R × R → R and o : R × R → R defined as a * b = | a – b | and a o b = a for all a, b $\in$ R. Show that '*' is commutative but not associative, 'o' is associative but not commutative.