The value of the integral int limits_{-log _{ | vedclass

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

Question

(D) Let $I = \int \limits_{-\ln 2}^{\ln 2} e^x \ln \left(e^x+\sqrt{1+e^{2 x}}\right) d x$.Substitute $e^x = t$,then $e^x dx = dt$. When $x = -\ln 2$,$

Vedclass · Accepted Answer

$\log _e\left(\frac{\sqrt{2}(2+\sqrt{5})^2}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}$

Vedclass · Answer

(D) Let $I = \int \limits_{-\ln 2}^{\ln 2} e^x \ln \left(e^x+\sqrt{1+e^{2 x}}\right) d x$.Substitute $e^x = t$,then $e^x dx = dt$. When $x = -\ln 2$,$t = 1/2$. When $x = \ln 2$,$t = 2$.$I = \int \limits_{1/2}^{2} \ln \left(t+\sqrt{1+t^2}\right) dt$.Using integration by parts,$\int u dv = uv - \int v du$. Let $u = \ln(t+\sqrt{1+t^2})$ and $dv = dt$.Then $du = \frac{1}{t+\sqrt{1+t^2}} \left(1 + \frac{t}{\sqrt{1+t^2}}\right) dt = \frac{1}{\sqrt{1+t^2}} dt$.$I = [t \ln(t+\sqrt{1+t^2})]_{1/2}^{2} - \int \limits_{1/2}^{2} \frac{t}{\sqrt{1+t^2}} dt$.$I = [2 \ln(2+\sqrt{5}) - \frac{1}{2} \ln(\frac{1}{2} + \sqrt{1 + 1/4})] - [\sqrt{1+t^2}]_{1/2}^{2}$.$I = 2 \ln(2+\sqrt{5}) - \frac{1}{2} \ln(\frac{1+\sqrt{5}}{2}) - (\sqrt{5} - \sqrt{5/4})$.$I = \ln((2+\sqrt{5})^2) - \ln((\frac{1+\sqrt{5}}{2})^{1/2}) - \frac{\sqrt{5}}{2}$.$I = \ln \left( \frac{(2+\sqrt{5})^2}{\sqrt{\frac{1+\sqrt{5}}{2}}} \right) - \frac{\sqrt{5}}{2} = \ln \left( \frac{\sqrt{2}(2+\sqrt{5})^2}{\sqrt{1+\sqrt{5}}} \right) - \frac{\sqrt{5}}{2}$.

The value of the integral $\int \limits_{-\log _{e} 2}^{\log _e 2} e^x \ln \left(e^x+\sqrt{1+e^{2 x}}\right) d x$ is equal to

Similar Questions

$\int_{0}^{\pi /2} \{ x - [\sin x] \} \,dx$ is equal to

The value of $b > 3$ for which $12 \int \limits_{3}^{b} \frac{1}{(x^{2}-1)(x^{2}-4)} dx = \log _{e}(\frac{49}{40})$ is equal to

$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot ^9 x \, dx =$

$\int_0^\pi \frac{1}{1+\sin x} dx$ is equal to

Consider the equation $\int_1^e \frac{(\log_e x)^{1/2}}{x(a-(\log_e x)^{3/2})^2} dx = 1$,where $a \in (-\infty, 0) \cup (1, \infty)$. Which of the following statements is/are $TRUE$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module