int sqrt{4 cos ^2 x - 5 sin ^2 x} cos x , dx | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $I = \int \sqrt{4 \cos ^2 x - 5 \sin ^2 x} \cos x \, dx$. Using the identity $\cos ^2 x = 1 - \sin ^2 x$, we get: $I = \int \sqrt{4(1 - \sin ^

Vedclass · Accepted Answer

$\frac{1}{2} \sin x \sqrt{4 - 9 \sin ^2 x} + \frac{2}{3} \sin ^{-1}\left(\frac{3 \sin x}{2}\right) + c$

Vedclass · Answer

(A) Let $I = \int \sqrt{4 \cos ^2 x - 5 \sin ^2 x} \cos x \, dx$. Using the identity $\cos ^2 x = 1 - \sin ^2 x$, we get: $I = \int \sqrt{4(1 - \sin ^2 x) - 5 \sin ^2 x} \cos x \, dx = \int \sqrt{4 - 9 \sin ^2 x} \cos x \, dx$. Let $3 \sin x = t$. Then $3 \cos x \, dx = dt$, which implies $\cos x \, dx = \frac{1}{3} dt$. Substituting these into the integral: $I = \int \sqrt{4 - t^2} \cdot \frac{1}{3} dt = \frac{1}{3} \int \sqrt{2^2 - t^2} dt$. Using the standard formula $\int \sqrt{a^2 - x^2} dx = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin ^{-1}\left(\frac{x}{a}\right) + c$: $I = \frac{1}{3} \left[ \frac{t}{2} \sqrt{4 - t^2} + \frac{4}{2} \sin ^{-1}\left(\frac{t}{2}\right) \right] + c$. Substituting $t = 3 \sin x$ back: $I = \frac{1}{3} \left[ \frac{3 \sin x}{2} \sqrt{4 - 9 \sin ^2 x} + 2 \sin ^{-1}\left(\frac{3 \sin x}{2}\right) \right] + c = \frac{1}{2} \sin x \sqrt{4 - 9 \sin ^2 x} + \frac{2}{3} \sin ^{-1}\left(\frac{3 \sin x}{2}\right) + c$.

$\int \sqrt{4 \cos ^2 x - 5 \sin ^2 x} \cos x \, dx =$

Similar Questions

If the roots of the given equation $(\cos p - 1){x^2} + (\cos p)x + \sin p = 0$ are real,then

$y=\log \left\{\left(\frac{1+x}{1-x}\right)^{1 / 4}\right\}-\frac{1}{2} \tan ^{-1}(x)$, then $\frac{d y}{d x}$ is equal to

If $x^2+px+1$ is a factor of the expression $ax^3+bx+c$,then

An electron accelerated through a potential difference $V$ passes through a uniform transverse magnetic field and experiences a force $F$. If the accelerating potential is increased to $2V$,the electron in the same magnetic field will experience a force:

Statement $S$: Presence of a rigid cell wall and nucleoprotein as genetic material.
Statement $R$: Examples include $Euglena$,$Dinoflagellates$,$Amoeba$,and $Paramecium$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module