The value of int e^{sin x} sin 2 x , dx is | vedclass

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

Question

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

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) We have,$I = \int e^{\sin x} \sin 2x \, dx$ Using the identity $\sin 2x = 2 \sin x \cos x$,we get: $I = \int e^{\sin x} (2 \sin x \cos x) \, dx = 2 \int e^{\sin x} \sin x \cos x \, dx$ Let $\sin x = t$,then $\cos x \, dx = dt$. Substituting these into the integral: $I = 2 \int e^t \cdot t \, dt$ 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: $I = 2 e^{\sin x} (\sin x - 1) + C$

The value of $\int e^{\sin x} \sin 2 x \, dx$ is

Similar Questions

$\int {\left[ {\frac{1}{{\log x}} - \frac{1}{{{{(\log x)}^2}}}} \right]dx = } $

If $\int x^3 \sin 3x \, dx = f(x) \cos 3x + g(x) \sin 3x + c$, then $27(f(x) + x g(x)) =$

If $\int(\log x)^3 x^5 d x=\frac{x^6}{A}\left[B(\log x)^3+C(\log x)^2+D(\log x)-1\right]+k$ and $A, B, C, D$ are integers, then $A-(B+C+D)=$

If $I_n = \int (\log x)^n \, dx$,then $I_n + n I_{n-1} =$

If $\int {x^5 e^{-x^2} dx} = g(x) e^{-x^2} + c$,where $c$ is a constant of integration,then $g(-1)$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module