int e^x cos x , dx = | vedclass

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

Question

(A) Let $I = \int e^x \cos x \, dx$. Using integration by parts,$\int u \, dv = uv - \int v \, du$. Let $u = \cos x$ and $dv = e^x \, dx$. Then $du =

Vedclass · Accepted Answer

$\frac{e^x (\cos x + \sin x)}{2} + c$

Vedclass · Answer

(A) Let $I = \int e^x \cos x \, dx$. Using integration by parts,$\int u \, dv = uv - \int v \, du$. Let $u = \cos x$ and $dv = e^x \, dx$. Then $du = -\sin x \, dx$ and $v = e^x$. $I = e^x \cos x - \int e^x (-\sin x) \, dx = e^x \cos x + \int e^x \sin x \, dx$. Now,evaluate $\int e^x \sin x \, dx$ using integration by parts again. Let $u = \sin x$ and $dv = e^x \, dx$. Then $du = \cos x \, dx$ and $v = e^x$. $\int e^x \sin x \, dx = e^x \sin x - \int e^x \cos x \, dx = e^x \sin x - I$. Substituting this back into the equation for $I$: $I = e^x \cos x + e^x \sin x - I$. $2I = e^x (\cos x + \sin x)$. $I = \frac{e^x (\cos x + \sin x)}{2} + c$.

$\int e^x \cos x \, dx =$

Similar Questions

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

$\int \sin ^{-1} x \, dx =$

Integrate the function: $\left(x^{2}+1\right) \log x$

$\int \log(x + 1) \, dx = $

$\int x \cos x \, dx = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module