Evaluate: (e^ x a +e^ a x +e^ a a ) d x - Vidyadip

Question

Evaluate: $\int\left(e^{x \log a}+e^{a \log x}+e^{a \log a}\right) d x$

✓

Answer

$\text { (c) : Let } I=\int\left(e^{x \log a}+e^{a \log x}+e^{a \log a}\right) d x$
$=\int\left(e^{\log a^x}+e^{\log x^a}+e^{\log a^a}\right) d x=\int\left(a^x+x^a+a^a\right) d x$
$=\frac{a^x}{\log a}+\frac{x^{a+1}}{a+1}+a^a x+c \quad\left[\because e^{\log y=y]}\right]$

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 "M.C.Q (1 Marks)" 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

The whole area of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ is :

If $\vec{\text{a}}$ is any vector, then $\big(\vec{\text{a}}\times\hat{\text{i}}\big)^2+\big(\vec{\text{a}}\times\hat{\text{j}}\big)^2+\big(\vec{\text{a}}\times\hat{\text{k}}\big)^2=$

$\vec{\text{a}}^2$
$2\vec{\text{a}}^2$
$3\vec{\text{a}}^2$
$4\vec{\text{a}}^2$

A cylindrical tank of radius 10m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of:

$1\text{m}/\text{hr}$
$0.1\text{m}/\text{hr}$
$1.1\text{m}/\text{hr}$
$0.5\text{m}/\text{hr}$

The solution of the differention $\frac{\text{dy}}{\text{dx}}+1=\text{e}^{\text{x}+\text{y}}$ is:

$(\text{x}+\text{y})\text{e}^{\text{x}+\text{y}}=0$
$(\text{x}+\text{C})\text{e}^{\text{x}+\text{y}}=0$
$(\text{x}-\text{C})\text{e}^{\text{x}+\text{y}}=1$
$(\text{x}-\text{C})\text{e}^{\text{x}+\text{y}}+1=0$

While measuring the side of an equilateral triangle an error of k % is made, the percentage error in its area is:

k%
2k%
$\frac{\text{K}}{2}\%$
3k%

The value of the integral $\int\limits^\infty_0\frac{\text{x}}{(1+\text{x})(1+\text{x}^2)}\text{dx}$ is:

$\frac{\pi}{2}$
$\frac{\pi}{4}$
$\frac{\pi}{6}$
$\frac{\pi}{3}$

In equation $3\text{x}-\text{y}\geq3$ and 4x - 4y > 4.

If $\vec{a} \times \vec{b}=\hat{i}-\hat{j}$ then the value of $\vec{b} \times \vec{a}$

The area bounded by the curve $\text{y}=\log_{\text{e}}\text{x}$ and x-axis and the straight line x = e is:

$\text{e sq. units}$
$1\text{ sq. units}$
$1-\frac{1}{\text{e}}\text{ sq. units}$
$1+\frac{1}{\text{e}}\text{ sq. units}$

If the area bounded by the curve $2 x^2+y^2=2$ is $A$. Then which of the following is the value of $A$ ?