Evalute the following integrals: e^ x-1 +x^ e-1 e^x+x^e dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{\text{e}^{\text{x}-1}+\text{x}^{\text{e}-1}}{\text{e}^\text{x}+\text{x}^\text{e}}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{\text{e}^{\text{x}-1}+\text{x}^{\text{e}-1}}{\text{e}^\text{x}+\text{x}^\text{e}}\text{dx}$
Putting $e^x + x^e = t$
$\Rightarrow\text{e}^\text{x}+\text{ex}^{\text{e}-1}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\text{e}\big(\text{e}^{\text{x}-1}+\text{x}^{\text{e}-1}\big)=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\big(\text{e}^{\text{x}-1}+\text{x}^{\text{e}-1}\big)\text{dx}=\frac{\text{dt}}{\text{e}}$
$\therefore\text{I}=\frac{1}{\text{e}}\int\frac{1}{\text{t}}\text{dt}$
$=\frac{1}{\text{e}}\text{ ln}|\text{t}|+\text{C}$
$=\frac{1}{\text{e}}\text{ ln}\Big|\text{e}^\text{x}+\text{x}^\text{e}\Big|+\text{C}\big[\because\text{t}=\text{e}^\text{x}+\text{x}^\text{e}\big]$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from INTEGRALS→INTEGRALS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

If $x=\sqrt{a^{\sin ^{-1} t}}$ and $y=\sqrt{a^{\cos ^{-1} t}}$ then prove that $\frac{d y}{d x}=-\frac{y}{x}$.

If $A = [a_{ij}]$ is a skew-symmetric matrix, then write the value of $\sum_\text{i}\text{a}_\text{ij}.$

Evaluate the following integrals:
$\int\sec^6\text{x }\tan\text{x}\text{ dx}$

In a factory, which manufactures nuts, machines A, B and C manufacture respectively 25%, 35% and 40% of nuts. Of their outputs, 5, 4 and 2 per cent respectively are defective nuts. A nut is drawn at random from the product and is found to be defective. Find the probability that it is manufactured by machine B.

Evaluate the following integrals:
$\int\frac{\text{x}^{\frac{-1}{3}}+\sqrt{\text{x}}+2}{\sqrt[3]{\text{x}}}\text{dx}$

Evalute the following integrals:
$\int\frac{1}{\sqrt{1-\text{x}^2}(2+3\sin^{-1}\text{x})}\text{dx}$

If $x ^{ y }= e ^{ x - y }$, then prove that $\frac{d y}{d x}=\frac{\log x}{(1+\log x)^2}$.

Show that the line through points $(1, -1, 2)$ and $(3, 4, -2)$ is perpendicular to the line throught the points $(0, 3, 2)$ and $(3, 5, 6).$

Integrate the following integrals:
$\int\cos3\text{x}\cos4\text{x dx}$

Find which of the function:
$\text{f(x)}=\begin{cases}\frac{|\text{x}-4|}{2(\text{x}-4)},&\text{if x}\neq4\\0,&\text{if x}=4\end{cases}$
at x = 4