If int frac{7x^8+8x^7}{(1+x+x^8)^2} dx = f(x) | vedclass

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

Question

(A) To find $f(x)$,we differentiate the given options to see which one yields the integrand $\frac{7x^8+8x^7}{(1+x+x^8)^2}$.Let $f(x) = \frac{x^8}{1+x

Vedclass · Accepted Answer

$\frac{x^8}{1+x+x^8}$

Vedclass · Answer

(A) To find $f(x)$,we differentiate the given options to see which one yields the integrand $\frac{7x^8+8x^7}{(1+x+x^8)^2}$.Let $f(x) = \frac{x^8}{1+x+x^8}$.Using the quotient rule $\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v u' - u v'}{v^2}$,where $u = x^8$ and $v = 1+x+x^8$:$f'(x) = \frac{(1+x+x^8)(8x^7) - (x^8)(1+8x^7)}{(1+x+x^8)^2}$$f'(x) = \frac{8x^7 + 8x^8 + 8x^{15} - x^8 - 8x^{15}}{(1+x+x^8)^2}$$f'(x) = \frac{7x^8 + 8x^7}{(1+x+x^8)^2}$Since the derivative of option $A$ matches the integrand,$f(x) = \frac{x^8}{1+x+x^8}$.

If $\int \frac{7x^8+8x^7}{(1+x+x^8)^2} dx = f(x) + c$,then $f(x)$ is equal to

Similar Questions

What are $X$ and $Y$ in the following reaction sequence?
$C_2H_5OH$ $\xrightarrow{Cl_2} \underline{X}$ $\xrightarrow{Cl_2} \underline{Y}$

The total charge induced in a conducting loop when it is moved in a magnetic field depends on

If $\tan \theta + \cot \theta = 2$,then $\sin \theta$ is equal to

The chloride of an element gives a neutral solution in water. To which group of the periodic table does this element belong?

An open pipe of length $33 \ cm$ resonates to a frequency of $1000 \ Hz$. The mode of vibration is: (velocity of sound $= 330 \ m/s$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module