int 2^x (f^{prime}(x) + f(x) log 2) , dx is e | vedclass

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

Question

(B) Let $I = \int 2^x (f^{\prime}(x) + f(x) \log 2) \, dx$. We can rewrite the integrand as: $I = \int (2^x f^{\prime}(x) + 2^x \log 2 \cdot f(x)) \,

Vedclass · Accepted Answer

$2^x f(x) + C$

Vedclass · Answer

(B) Let $I = \int 2^x (f^{\prime}(x) + f(x) \log 2) \, dx$. We can rewrite the integrand as: $I = \int (2^x f^{\prime}(x) + 2^x \log 2 \cdot f(x)) \, dx$. Recall the product rule for differentiation: $\frac{d}{dx} [u(x)v(x)] = u(x)v^{\prime}(x) + v(x)u^{\prime}(x)$. Let $u(x) = 2^x$ and $v(x) = f(x)$. Then $u^{\prime}(x) = \frac{d}{dx} (2^x) = 2^x \log 2$. Thus, the integrand is of the form $u(x)v^{\prime}(x) + v(x)u^{\prime}(x)$, which is $\frac{d}{dx} [2^x f(x)]$. Therefore, $I = \int \frac{d}{dx} [2^x f(x)] \, dx = 2^x f(x) + C$.

$\int 2^x (f^{\prime}(x) + f(x) \log 2) \, dx$ is equal to:

Similar Questions

$\int \frac{(x-1) e^{x}}{(x+1)^{3}} d x$ is equal to

$\int e^{x}\left(\frac{1-x}{1+x^{2}}\right)^{2} \,d x=$

$\int e^x \frac{x^2+1}{(x+1)^2} d x$ is equal to

The value of $\int \frac{e^{\tan ^{-1}(x)}}{1+x^2} \left[\left(\sec ^{-1} \sqrt{1+x^2}\right)^2+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right] d x$,for $x>0$ is

$\int \frac{e^{\sin x}(\sin 2x - 8 \cos x)}{2(\sin x - 3)^2} dx =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module