int sin ^3(x) cdot cos ^3(x) , dx = | vedclass

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

Question

(C) We need to evaluate the integral $I = \int \sin ^3(x) \cdot \cos ^3(x) \, dx$. Rewrite the integral as: $I = \int \sin ^3(x) \cdot \cos ^2(x) \cdo

Vedclass · Accepted Answer

$\frac{1}{4} \sin ^4(x) - \frac{1}{6} \sin ^6(x) + C$

Vedclass · Answer

(C) We need to evaluate the integral $I = \int \sin ^3(x) \cdot \cos ^3(x) \, dx$. Rewrite the integral as: $I = \int \sin ^3(x) \cdot \cos ^2(x) \cdot \cos(x) \, dx$ Using the identity $\cos ^2(x) = 1 - \sin ^2(x)$,we get: $I = \int \sin ^3(x) \cdot (1 - \sin ^2(x)) \cdot \cos(x) \, dx$ Let $t = \sin(x)$,then $dt = \cos(x) \, dx$. Substituting these into the integral: $I = \int t^3(1 - t^2) \, dt = \int (t^3 - t^5) \, dt$ Integrating with respect to $t$: $I = \frac{t^4}{4} - \frac{t^6}{6} + C$ Substituting back $t = \sin(x)$: $I = \frac{1}{4} \sin ^4(x) - \frac{1}{6} \sin ^6(x) + C$

$\int \sin ^3(x) \cdot \cos ^3(x) \, dx =$

Similar Questions

Let $n \ge 2$ be a natural number and $0 < \theta < \frac{\pi}{2}$. Then $\int \frac{(\sin^n \theta - \sin \theta)^{\frac{1}{n}} \cos \theta}{\sin^{n+1} \theta} d\theta$ is equal to

$\int \frac{\cos x - \sin x}{\sqrt{\sin 2x}} \, dx$ equals

If $\int \frac{\sqrt{1-x^2}}{x^4} \,dx = A(x)\left(\sqrt{1-x^2}\right)^{m} + c$ for a suitably chosen integer $m$ and a function $A(x)$,where $c$ is a constant of integration,then $(A(x))^{m}$ equals

$\int {\frac{{dx}}{{{x^2}{{(1 + {x^4})}^{\frac{3}{4}}}}}} $ is equal to- (where $C$ is constant of integration)

$\int x^{2019} \cdot e^{x^{2020}} \, dx = $ . . . . . . $+ C$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module