Observe the following statements :A: int left | vedclass

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

Question

(C) For statement $A$: Let $I = \int \left(\frac{x^2-1}{x^2}\right) e^{\left(\frac{x^2+1}{x}\right)} d x$.We can rewrite the integrand as $I = \int \l

Vedclass · Accepted Answer

$A$ is true,$R$ is false

Vedclass · Answer

(C) For statement $A$: Let $I = \int \left(\frac{x^2-1}{x^2}\right) e^{\left(\frac{x^2+1}{x}\right)} d x$.We can rewrite the integrand as $I = \int \left(1 - \frac{1}{x^2}\right) e^{\left(x + \frac{1}{x}\right)} d x$.Let $t = x + \frac{1}{x}$. Then $dt = \left(1 - \frac{1}{x^2}\right) d x$.Substituting these into the integral,we get $I = \int e^t d t = e^t + c = e^{x + \frac{1}{x}} + c = e^{\frac{x^2+1}{x}} + c$. Thus,statement $A$ is true.For statement $R$: The integral $\int f^{\prime}(x) e^{f(x)} d x$ is evaluated by substituting $t = f(x)$,which gives $dt = f^{\prime}(x) d x$.Thus,$\int e^t d t = e^t + c = e^{f(x)} + c$.The statement $R$ claims the result is $f(x) + c$,which is incorrect.Therefore,$A$ is true and $R$ is false.

Observe the following statements :
$A: \int \left(\frac{x^2-1}{x^2}\right) e^{\frac{x^2+1}{x}} d x = e^{\frac{x^2+1}{x}} + c$
$R: \int f^{\prime}(x) e^{f(x)} d x = f(x) + c$
Then which of the following is true?

Similar Questions

If $\int \frac{dx}{2 \cos x + 3 \sin x + 4} = \frac{2}{\sqrt{3}} f(x) + c$,then $f\left(\frac{2 \pi}{3}\right) =$

For $n \ge 2$,if $I_n = \int (\sin x + \cos x)^n dx$,then $nI_n - 2(n-1)I_{n-2} = $

$\int \frac{x - 1}{(x + 1) \sqrt{x(x^2 + x + 1)}} dx =$

The value of $\frac{\int_0^{\pi / 2}(\sin x)^{\sqrt{2}+1} d x}{\int_0^{\pi / 2}(\sin x)^{\sqrt{2}-1} d x}$ is $........$

$ \int \frac{1}{x^{2}\left(x^{4}+1\right)^{3 / 4}} dx $ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Observe the following statements :$A: \int \left(\frac{x^2-1}{x^2}\right) e^{\frac{x^2+1}{x}} d x = e^{\frac{x^2+1}{x}} + c$$R: \int f^{\prime}(x) e^{f(x)} d x = f(x) + c$Then which of the following is true?