Evaluate the following integrals: ^ 2 _0 ( + )dx - Vidyadip

Question

Evaluate the following integrals:$\int\limits^{2\pi}_0\log(\sec\text{x}+\tan\text{x})\text{dx}$

✓

Answer

We know,$\int\limits^{2\pi}_0\text{f(x)}\text{dx}=\int\limits^{2\pi}_0\text{f}(2\pi-\text{x})\text{dx}$
Hence,$\int\limits^{2\pi}_0\log(\sec\text{x}+\tan\text{x})\text{dx}=\int\limits^{2\pi}_0\log\big(\sec(2\pi-\text{x})+\tan(2\pi-\text{x})\big)\text{dx}$
$\int\limits^{2\pi}_0\log(\sec\text{x}+\tan\text{x})\text{dx}=\int\limits^{2\pi}_0\log(\sec\text{x}-\tan\text{x})\text{dx}$
If$\text{I}=\int\limits^{2\pi}_0\log(\sec\text{x}+\tan\text{x})\text{dx}$
$2\text{I}=\int\limits^{2\pi}_0\log(\sec\text{x}+\tan\text{x})\text{dx}+\int\limits^{2\pi}_0\log(\sec\text{x}-\tan\text{x})\text{dx}$
$2\text{I}=\int\limits^{2\pi}_0\log(\sec\text{x}+\tan\text{x})\text{dx}+\log(\sec\text{x}-\tan\text{x})\text{dx}$
$2\text{I}=\int\limits^{2\pi}_0\log(\sec^2\text{x}-\tan^2\text{x})\text{dx}$
$2\text{I}=\int\limits^{2\pi}_0\log(1)\text{dx}$
$2\text{I}=0$
$\text{I}=0$

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 "Solve the Following Question.(5 Marks)" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If a, b, c are the langths of sides, BC, CA and AB of a triangle ABC, prove that $\overrightarrow{\text{BC}}+\overrightarrow{\text{CA}}+\overrightarrow{\text{AB}}=\vec{\text{0}}$ and deduce that $\frac{\text{a}}{\sin\text{A}}=\frac{\text{b}}{\sin\text{B}}=\frac{\text{c}}{\sin\text{C}}.$

$\int\frac{\text{x}^2+\text{x}+5}{3\text{x}+2}\text{dx}$

Differentiate the following functions with respect to x:
$\text{e}^{\text{ax}}\sec\text{x}\tan2\text{x}$

Give examples of two surjective functions $f_1$ and $f_2$ from $Z$ to $Z$ such that $f_1 + f_2$ is not surjective.

Discuss the continuity of the following functions at the indicated point:
$\text{f}\text{(x)}=\begin{cases}|\text{x}-\text{a|}\sin(\frac{1}{\text{x}-\text{a}}), &\text{for} \text{ x} \neq\text{a}\\0,&\text{for} \text{ x} = \text{a}\end{cases}\text{ at x}=0$

In a simple circult of resistance R, self inductance L and voltage E, the current i at any times t is given by $\text{L}\frac{\text{di}}{\text{dt}}+\text{R}\text{i}=\text{E}.$ If E is constant and initially no current throught the circuit, prove that $\text{i}=\frac{\text{E}}{\text{R}}\left\{1-\text{e}^-(\frac{\text{R}}{\text{L}})\text{t}\right\}.$

Show that the vectors
$\vec{\text{a}}=\frac{1}{7}(2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}),\vec{\text{b}}=\frac{1}{7}(3\hat{\text{i}}-6\hat{\text{j}}+2\hat{\text{k}}),\vec{\text{c}}=\frac{1}{7}(6\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}})$ are mutually perpendicular unit vectors.

Evaluate the following determinant:
$\begin{vmatrix}6&-3&2\\2&-1&2\\-10&5&2 \end{vmatrix}$

Evaluate the following integrals as limit of sum:$\int\limits^{4}_{0}\big(\text{x}+\text{e}^{2\text{x}}\big)\text{dx}$

Show that the following system of linear equations is consistent and also find solution:
$2x + 3y = 5$
$6x + 9y = 15$