int {e^{x/2}} sin left( frac{x}{2} + frac{pi} | vedclass

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

Question

(D) Let $I = \int {e^{x/2}} \sin \left( \frac{x}{2} + \frac{\pi}{4} \right) \, dx$. Using the formula $\int e^{ax} \sin(bx + c) \, dx = \frac{e^{ax}}{

Vedclass · Accepted Answer

$\sqrt{2} {e^{x/2}} \sin \frac{x}{2} + c$

Vedclass · Answer

(D) Let $I = \int {e^{x/2}} \sin \left( \frac{x}{2} + \frac{\pi}{4} \right) \, dx$. Using the formula $\int e^{ax} \sin(bx + c) \, dx = \frac{e^{ax}}{a^2 + b^2} [a \sin(bx + c) - b \cos(bx + c)] + C$. Here $a = 1/2$ and $b = 1/2$. $I = \frac{e^{x/2}}{(1/2)^2 + (1/2)^2} \left[ \frac{1}{2} \sin \left( \frac{x}{2} + \frac{\pi}{4} \right) - \frac{1}{2} \cos \left( \frac{x}{2} + \frac{\pi}{4} \right) \right] + C$. $I = \frac{e^{x/2}}{1/4 + 1/4} \cdot \frac{1}{2} \left[ \sin \left( \frac{x}{2} + \frac{\pi}{4} \right) - \cos \left( \frac{x}{2} + \frac{\pi}{4} \right) \right] + C$. $I = \frac{e^{x/2}}{1/2} \cdot \frac{1}{2} \left[ \sin \left( \frac{x}{2} + \frac{\pi}{4} \right) - \cos \left( \frac{x}{2} + \frac{\pi}{4} \right) \right] + C$. $I = e^{x/2} \left[ \sin \frac{x}{2} \cos \frac{\pi}{4} + \cos \frac{x}{2} \sin \frac{\pi}{4} - (\cos \frac{x}{2} \cos \frac{\pi}{4} - \sin \frac{x}{2} \sin \frac{\pi}{4}) \right] + C$. Since $\sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}$,$I = e^{x/2} \left[ \frac{1}{\sqrt{2}} \sin \frac{x}{2} + \frac{1}{\sqrt{2}} \cos \frac{x}{2} - \frac{1}{\sqrt{2}} \cos \frac{x}{2} + \frac{1}{\sqrt{2}} \sin \frac{x}{2} \right] + C$. $I = e^{x/2} \left[ \frac{2}{\sqrt{2}} \sin \frac{x}{2} \right] + C = \sqrt{2} {e^{x/2}} \sin \frac{x}{2} + C$. Thus,the correct option is $D$.

$\int {e^{x/2}} \sin \left( \frac{x}{2} + \frac{\pi}{4} \right) \, dx = $

Similar Questions

If $\int \frac{\sqrt{1-x^4}}{x^7} d x=f(x)\left\{\sqrt{1-x^4}\right\}^n+C$,then $(f(x))^n$ is equal to

$\int \sqrt{x-1}(x \sqrt{x+1})^{-1} d x=$

If $\int \frac{a \cos x+3 \sin x}{5 \cos x+2 \sin x} d x=\frac{26}{29} x-\frac{k}{29} \log |5 \cos x+2 \sin x|+c$,then $|a+k|=$

If $\int {\frac{{dx}}{{{x^3}{{\left( {1 + {x^6}} \right)}^{2/3}}}} = xf\left( x \right){{\left( {1 + {x^6}} \right)}^{\frac{1}{3}}} + C} $ where $C$ is a constant of integration,then the function $f(x)$ is equal to

Let $I(x) = \int \frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If $I(37) - I(24) = \frac{1}{4} \left( \frac{1}{b^{\frac{1}{13}}} - \frac{1}{c^{\frac{1}{13}}} \right)$,where $b, c \in \mathbb{N}$,then $3(b+c)$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module