int e^{sin x} sin 2x , dx = . . . . . . + | vedclass

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

Question

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

Vedclass · Accepted Answer

$2e^{\sin x}(\sin x - 1)$

Vedclass · Answer

(A) Let $I = \int e^{\sin x} \sin 2x \, dx$. Using the identity $\sin 2x = 2 \sin x \cos x$,we get: $I = \int e^{\sin x} (2 \sin x \cos x) \, dx$. Let $t = \sin x$,then $dt = \cos x \, dx$. Substituting these into the integral: $I = \int e^t (2t) \, dt = 2 \int t e^t \, dt$. Using integration by parts,$\int u \, dv = uv - \int v \, du$,where $u = t$ and $dv = e^t \, dt$: $I = 2 [t e^t - \int e^t \, dt] = 2 [t e^t - e^t] + c$. $I = 2 e^t (t - 1) + c$. Substituting $t = \sin x$ back: $I = 2 e^{\sin x} (\sin x - 1) + c$. Thus,the correct option is $A$.

$\int e^{\sin x} \sin 2x \, dx = $ . . . . . . $+ c$.

Similar Questions

If $\int f(x) dx = \psi(x)$,then $\int x^5 f(x^3) dx = $

If $\int x^3 e^{5 x} d x = \frac{e^{5 x}}{5^4}[f(x)] + C$,then $f(x)$ is equal to

If $f(x) = \lim _{n \rightarrow \infty} n^2 \left(x^{\frac{1}{n}} - x^{\frac{1}{n+1}}\right), x > 0$,then $\int x f(x) d x =$

$\int {32{x^3}{{(\log x)}^2}dx} $ is equal to

If $n \in N$ and $I_{n}=\int(\log x)^{n} dx$,then $I_{n}+n I_{n-1}$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module