Evaluate the following integrals: 4 ^2x-1 dx - Vidyadip

Question

Evaluate the following integrals:$\int\frac{\sin\text{x}}{\sqrt{4\cos^2\text{x}-1}}\text{ dx}$

✓

Answer

Let $\text{I}=\int\frac{\sin\text{x}}{\sqrt{4\cos^2\text{x}-1}}\text{ dx}$
Let $2\cos\text{x}=\text{t}$
$\Rightarrow-2\sin\text{x}\text{ dx}=\text{dt}$
$\Rightarrow\sin\text{x}\text{ dx}=-\frac{\text{dt}}{2}$
$\text{I}=-\frac{1}{2}\int\frac{\text{dt}}{\sqrt{\text{t}^2-1}}$
$=-\frac{1}{2}\log\Big|\text{t}+\sqrt{\text{t}^2-1}\Big|+\text{C}$ $\Big[\text{Since }\int\frac{1}{\sqrt{\text{x}^2-\text{a}^2}}\text{ dx}=\log\Big|\text{x}+\sqrt{\text{x}^2+\text{a}^2}\Big|+\text{C}\Big]$
$=-\frac{1}{2}\log\Big|2\cos\text{x}+\sqrt{4\cos^2\text{x}-1}\Big|+\text{C}$

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 "Solve the Following Question.(3 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the equation of the plane which passes through the point (3, 4, -1) and is parallel to the plane 2x - 3y + 5z + 7 = 0. Also, find the distance between the two planes.

If $\text{y}=\tan^{-1}$ show that $(1+\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}+2\text{x}\frac{\text{dy}}{\text{dx}}=0$

Find $p$ and $q$. If the equation $p x^2-8 x y+3 y^2+14 x+2 y+q=0$ represents a pair of perpendicular lines.

A can hit a target 3 times in 6 shots, B : 2 times in 6 shots and C : 4 times in 4 shots. They fix a volley. What is the probability that at least 2 shots hit?

Diffrentiate the following w.r.t.x

$\frac{1+\sin x^{\circ}}{1-\sin x^{\circ}}$

If the origin is the centroid of the triangle whose vertices are $A (2, p,-3), B (q,-2,5)$ and $R (-5,1, r)$, then find the values of $p, q, r$.

Evaluate the following integrals:$\int\frac{1}{\text{x}\sqrt{4-9(\log\text{x})^2}}\text{ dx}$

Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers. Find E(X).

A circular disc of radius 3cm is being heated. Due to expansion, its radius increases at the rate of 0.05cm/ sec. Find the rate at which its area is increasing when radius is 3.2cm.

Write a value of $\int\frac{(\log\text{x})^{\text{n}}}{\text{x}}\text{ dx}$