The value of the definite integral int_{0}^{1 | vedclass

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

Question

(A) Let $I = \int_{0}^{1} e^{e^x}(1 + x e^x) dx$. Consider the substitution $u = x e^x$. Then $du = (1 \cdot e^x + x \cdot e^x) dx = e^x(1 + x) dx$. T

Vedclass · Accepted Answer

$e^e$

Vedclass · Answer

(A) Let $I = \int_{0}^{1} e^{e^x}(1 + x e^x) dx$. Consider the substitution $u = x e^x$. Then $du = (1 \cdot e^x + x \cdot e^x) dx = e^x(1 + x) dx$. This does not simplify directly. Let us rewrite the integral as $I = \int_{0}^{1} e^{e^x} dx + \int_{0}^{1} x e^x e^{e^x} dx$. Consider the second part: $\int_{0}^{1} x e^x e^{e^x} dx$. Let $v = e^x$. Then $dv = e^x dx$. When $x=0, v=1$. When $x=1, v=e$. The integral becomes $\int_{1}^{e} \ln(v) e^v dv$. This is not standard. Let us re-examine the original expression: $\int_{0}^{1} e^{e^x} dx + \int_{0}^{1} x e^x e^{e^x} dx$. Actually,notice that $\frac{d}{dx}(x e^{e^x}) = 1 \cdot e^{e^x} + x \cdot e^{e^x} \cdot e^x = e^{e^x} + x e^x e^{e^x} = e^{e^x}(1 + x e^x)$. Thus,the integral is simply $\left[ x e^{e^x} \right]_{0}^{1}$. Evaluating at the limits: $(1 \cdot e^{e^1}) - (0 \cdot e^{e^0}) = e^e - 0 = e^e$.

The value of the definite integral $\int_{0}^{1} e^{e^x}(1 + x e^x) dx$ is equal to

Similar Questions

Number of ordered pairs $(a, b)$ satisfying simultaneously the system of equations $\int_{a}^{b} x^{3} dx = 0$ and $\int_{a}^{b} x^{2} dx = \frac{2}{3}$ is

Evaluate the definite integral $\int_{-1}^{1}(x+1) d x$.

$\int_0^{\pi /4} (\cos x - \sin x) dx + \int_{\pi /4}^{5\pi /4} (\sin x - \cos x) dx + \int_{2\pi }^{\pi /4} (\cos x - \sin x) dx$ is equal to

If $f$ is integrable on $[0, a]$,then the function $h$ defined on $[0, a]$ as $h(x) = \int_0^x f(t) dt$ is integrable on $[0, a]$. Which of the following functions is also integrable on $[0, a]$?

If $f(x)=\sin ^6 x+\cos ^6 x+2 \sin ^3 x \cos ^3 x$, then $\int_0^{\pi / 4} \frac{\sin ^2 2 x}{f(x)} d x=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module