Evaluate the following integrals: 2x ^4x+4 ^2x-2 dx - Vidyadip

Question

Evaluate the following integrals:

$\int\frac{\sin2\text{x}}{\sqrt{\sin^4\text{x}+4\sin^2\text{x}-2}}\text{ dx}$

✓

Answer

$\int\frac{\sin(2\text{x})\text{dx}}{\sqrt{\sin^4\text{x}+4\sin^2\text{x}-2}}$
Let $\sin^2\text{x}=\text{t}$
$\Rightarrow2\sin\text{x}\cos\text{x}\text{ dx}=\text{dt}$
$\Rightarrow\sin(2\text{x})\text{ dx}=\text{dt}$
Now, $\int\frac{\sin(2\text{x})\text{dx}}{\sqrt{\sin^4\text{x}+4\sin^2\text{x}-2}}$
$=\int\frac{\text{dt}}{\sqrt{\text{t}^2+4\text{t}-2}}$
$=\int\frac{\text{dt}}{\sqrt{\text{t}^2+4\text{t}+4-4-2}}$
$=\int\frac{\text{dt}}{\sqrt{(\text{t}+2)^2-\big(\sqrt6\big)^2}}$
$=\log\Big|\text{t}+2+\sqrt{(\text{t}+2)^2-6}\Big|+\text{C}$
$=\log\Big|\text{t}+2+\sqrt{\text{t}^2+4\text{t}-2}\Big|+\text{C}$
$=\log\Big|\sin^2\text{x}+2\sqrt{\sin^4\text{x}+4\sin^2\text{x}-2}\Big|+\text{C}$

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 "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

Write the minors and cofactors of element of the first column of the following matrices and hence evaluate the determinant in case:
$\text{A}=\begin{vmatrix}2&-1&0&1\\-3&0&1&-2\\1&1&-1&1\\2&-1&5&0 \end{vmatrix}$

Find the particular solution of the differential equation(1 + y²) + (x – $\text{e}^{\tan^{-1}}$y)$\frac{\text{dy}}{\text{dx}}=0$ given that y = 0 when x=1.

Evaluate the following integrals:
$\int^\limits2_{0}\big|\text{x}^2-3\text{x}+2\big|\text{dx}$

R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x - 3. Then, R^-1 is:

{(8, 11), (10, 13)}
{(11, 8), (13, 10)}
{(10, 13), (8, 11)}
None of these.

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

Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x + y}}{\text{x}-\text{y}}$

How many packets of each food should be used to maximise the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet?

In order to supplement daily diet, a person wishes to take X and Y tablets. The contents (in milligrams per tablet) of iron, calcium and vitamins in X and Y are given as below:

Tablets	Iron	Calcium	Vitamin
X	6	3	2
Y	2	3	4

The person needs to supplement at least 18 milligrams of iron, 21 milligrams of calcium and 16 milligrams of vitamins. The price of each tablet of X and Y is ₹ 2 and ₹1 respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an LPP and solve graphically.

Evaluate the following integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\frac{1}{5+4\sin\text{x}}\text{ dx}$

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=(\tan\text{x})^{\cot\text{x}}+(\cot\text{x})^{\tan\text{x}}$