Evalute the following integrals: cosec x x 2 dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{\text{cosec x}}{\log\tan\frac{\text{x}}{2}}\text{dx}$

✓

Answer

Let $\int\frac{\text{cosec x}}{\log\tan\frac{\text{x}}{2}}\text{dx}\ .....(\text{i})$
Let $\log\tan\frac{\text{x}}{2}=\text{t}$ then,
$\text{d}\Big[\log\tan\frac{\text{x}}{2}\Big]=\text{dt}$
$\Rightarrow\text{cosec x dx}=\text{dt}$
$\Rightarrow\text{dx}=\frac{\text{dt}}{\text{cosec x}}$
Putting $\log\tan\frac{\text{x}}{2}=\text{t}$ and $\text{dx}=\frac{\text{dt}}{\text{cosec x}}$ in equation (i), we get
$\text{I}=\int\frac{\text{cosec x}}{\text{t}}\times\frac{\text{dt}}{\text{cosec x}}$
$=\int\frac{\text{dt}}{\text{t}}$
$=\log|\text{t}|+\text{C}$
$=\log\Big|\log\tan\frac{\text{x}}{2}\Big|+\text{C}$
$\therefore\text{I}=\log\Big|\log\tan\frac{\text{x}}{2}\Big|+\text{C}$

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 "5 Marks Questions" 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

The decay rate of radium at any time t is proportional to its mass at that time. Find the time when the mass will be halved of its intial mass.

Examine the differentiability of f, where f is defined by:
$\text{f(x)}=\begin{cases}1+\text{x},&\text{if x}\leq2\\5-\text{x},&\text{if x}>2\end{cases}$
at x = 2.

In the following, find the value of the constant k so that the given function is continuous at the indicated point:
$\text{f(x)}=\begin{cases}\frac{\text{x}^2+\text{x}^2-16\text{x}+20}{(\text{x}-2)^2},&\text{ x}\neq2\\\text{k},&\text{x}=2\end{cases}$

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

Prove the following Exercise:
$\int^{3}\limits_{1}\frac{\text{dx}}{\text{x}^{2}(\text{x}+1)}=\frac{2}{3}+\log\frac{2}{3}$

Let R be the relation over the set of all straight lines in a plane such that $\text{l}_1\text{Rl}_2\Leftrightarrow\text{l}_1\bot\text{l}_2.$ Then, R is:

Symmetric.
Reflexive.
Transitive.
An equivalence relation.

Solve the following system of equations by matrix method:
2x + y + z = 2
x + 3y − z = 5
3x + y − 2z = 6

Examine the continuity of the function
$\text{f}\text{(x)}=\begin{cases}3\text{x}-2 &, \text{ if x} \leq 0\\\text{x}+1 &, \text{ if x} > 0\end{cases}\text{at x}=0$
Also sketch the graph of this function.

Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.

If $\text{A}=\begin{bmatrix}-1 & 2 & 0 \\ -1 & 1 & 1 \\ 0 & 1 & 0 \end{bmatrix},$ show that $A^2 = A^{-1}.$