Evaluate: (a^x+b^x )^2 a^x b^x d x - Vidyadip

MCQ

Evaluate: $\int \frac{\left(a^x+b^x\right)^2}{a^x b^x} d x$

✓
$\frac{\left(\frac{a}{b}\right)^x}{\log \frac{a}{b}}+\frac{\left(\frac{b}{a}\right)^x}{\log \frac{b}{a}}+2 x+C, a \neq b$
B
$\frac{\left(\frac{a}{b}\right)^x}{\log \frac{a}{b}}+\frac{\left(\frac{b}{a}\right)^x}{\log \frac{a}{b}}+2 x+C, a \neq b$
C
$\left(\frac{a}{b}\right)^x+\left(\frac{b}{a}\right)^x+2 x+C, a \neq b$
D
None of these

✓

Answer

Correct option: A.

$\frac{\left(\frac{a}{b}\right)^x}{\log \frac{a}{b}}+\frac{\left(\frac{b}{a}\right)^x}{\log \frac{b}{a}}+2 x+C, a \neq b$

(a) : We have, $\int \frac{\left(a^x+b^x\right)^2}{a^x b^x} d x=\int \frac{a^{2 x}+b^{2 x}+2 a^x b^x}{a^x b^x} d x$
$
=\int\left(\left(\frac{a}{b}\right)^x+\left(\frac{b}{a}\right)^x+2\right) d x=\frac{\left(\frac{a}{b}\right)^x}{\log \frac{a}{b}}+\frac{\left(\frac{b}{a}\right)^x}{\log \frac{b}{a}}+2 x+C, a \neq b
$

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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $y = (1 + {x^2}){\tan ^{ - 1}}x - x,$ then ${{dy} \over {dx}} = $

The value of $\int_{\, - 2}^{\,2} {\left[ {p\ln \left( {\frac{{1 + x}}{{1 - x}}} \right) + q\ln {{\left( {\frac{{1 - x}}{{1 + x}}} \right)}^{ - 2}} + r} \right]\,dx} $ depends on

If $g(x)=x^2+x-2$ and $\frac{1}{2} \operatorname{gof}(x)=2 x^2-5 x+2$, then $f(x)$ is equal to:

The value of a so that the sum of the squares of the roots of the equation ${x^2} - (a - 2)x - a + 1 = 0$ assume the least value, is

If $A\, = \,\left[ \begin{gathered}
1\ \ \ \,1\ \ \ \,2\ \ \ \hfill \\
0\ \ \ \,2\ \ \ \,1\ \ \ \hfill \\
1\ \ \ \,0\ \ \ \,2\ \ \ \hfill \\
\end{gathered} \right]$ and $A^3 = (aA-I) (bA-I)$,where $a, b$ are integers and $I$ is a $3 × 3$ unit matrix then value of $(a + b)$ is equal to

The direction ratios of the line of intersection of the planes $3x + 2y - z = 5$ and $x - y + 2z = 3$ are:

$A$ and $B$ toss a fair coin each simultaneously $50$ times. The probability that both of them will not get tail at the same toss is

The vector equation of the line joining the points $i-2j+k$ and $-2j+3k$ is

Let $f:(0, \infty) \rightarrow R$ be given by $f(x)=\int_{\frac{1}{x}}^x e^{-\left(t+\frac{1}{t}\right)} \frac{d t}{t}$. Then

$(A)$ $f(x)$ is monotonically increasing on $[1, \infty)$

$(B)$ $f(x)$ is monotonically decreasing on $(0,1)$

$(C)$ $f(x)+f\left(\frac{1}{x}\right)=0$, for all $x \in(0, \infty)$

$(D)$ $f\left(2^x\right)$ is an odd function of $x$ on $R$

If the graph of non-constant function is symmetric about the point $(3,4)$ , then the value of $\sum\limits_{r = 0}^6 {f(r) + f(3)} $ is equal to