Evaluate the following integrals: ^ 6 _ 0 ^ -3 2 2 d - Vidyadip

Question

Evaluate the following integrals:
$\int^\limits{\frac{\pi}{6}}_{0}\cos^{-3}2\theta\sin2\theta\text{ d}\theta$

✓

Answer

We have,
$\int^\limits{\frac{\pi}{6}}_{0}\cos^{-3}2\theta\sin2\theta\text{ d}\theta$
$=\int^\limits{\frac{\pi}{6}}_{0}\frac{\sin2\theta}{\cos^32\theta}\text{ d}\theta$
$=\int^\limits{\frac{\pi}{6}}_{0}\tan2\theta\cdot\sec^22\theta\text{ d}\theta$
Let $\tan2\theta=\text{t}$
Differentiating w.r.t. x, we get
$2\sec^22\theta\text{d}\theta=\text{dt}$
Now, $\theta=0\Rightarrow\text{t}=0$
$\theta=\frac{\pi}{6}\Rightarrow\text{t}=\sqrt{3}$
$\therefore\ \int^\limits{\frac{\pi}{6}}_{0}\tan2\theta\cdot\sec^22\theta\text{ d}\theta=\frac{1}{2}\int^\limits{\sqrt{3}}_0\text{t dt}=\frac{1}{2}\Big[\frac{\text{t}^2}{2}\Big]^{\sqrt{\text{3}}}_0$
$=\frac{3}{4}$

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 INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Write the following in the simplest form:
$\tan^{-1}\sqrt{\frac{\text{x}}{\text{a}+\sqrt{\text{a}^2-\text{x}^2}}},-\text{a}<\text{x}<\text{a}$

The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices.

Discuss the continuity of the following functions:

$\text{f(x)}=\sin\text{x}+\cos\text{x}$
$\text{f(x)}=\sin\text{x}-\cos\text{x}$
$\text{f(x)}=\sin\text{x}\cos\text{x}$

A box of constant volume c is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions?

Show that the line segments joining the mid-points of opposite sides of a quadrilateral bisects each other.

If f is defined by $\text{f(x)}=\text{x}^2-4\text{x}+7,$ show that $\text{f}'(5)=2\text{f}'\Big(\frac{7}{2}\Big).$

Evaluate the following integrals:
$\int4\text{x}^3\sqrt{5-\text{x}^2}\text{ dx}$

Using integration find the area of the region bounded by the curves $\text{y} = \sqrt{4 - \text{x}^{2}}, \text{x}^{2} + \text{y}^{2} \text{4x} = 0$ and the x-axis.

Find the intervals in which the following functions are increasing or decreasing.
f(x) = (x - 1)(x - 2)²

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\frac{\text{x}^2}{\text{a}^2}-\frac{\text{y}^2}{\text{b}^2}=1\text{ at }(\text{x}_1,\text{y}_1)$