_ ^ dx x( x^7 + 1) = - Vidyadip

MCQ

$\int_{}^{} {\frac{{dx}}{{x({x^7} + 1)}}} = $

A
$\log \left( {\frac{{{x^7}}}{{{x^7} + 1}}} \right) + c$
✓
$\frac{1}{7}\log \left( {\frac{{{x^7}}}{{{x^7} + 1}}} \right) + c$
C
$\log \left( {\frac{{{x^7} + 1}}{{{x^7}}}} \right) + c$
D
$\frac{1}{7}\log \left( {\frac{{{x^7} + 1}}{{{x^7}}}} \right) + c$

✓

Answer

Correct option: B.

$\frac{1}{7}\log \left( {\frac{{{x^7}}}{{{x^7} + 1}}} \right) + c$

b
(b) Given, $\int_{}^{} {\frac{{dx}}{{x\,({x^7} + 1)}}} = \int_{}^{} {\frac{{dx}}{{{x^8}\left( {1 + \frac{1}{{{x^7}}}} \right)}}} $
Put $1 + \frac{1}{{{x^7}}} = t$ ==> $\frac{{ - 7}}{{{x^8}}}dx = dt$
$I = \frac{{ - 1}}{7}\int {\frac{{dt}}{t} = } \frac{{ - 1}}{7}\log t + c$
==> $I = - \frac{1}{7}\log \left( {\frac{{{x^7} + 1}}{{{x^7}}}} \right) + c$ ==> $I = \frac{1}{7}\log \left( {\frac{{{x^7}}}{{{x^7} + 1}}} \right) + c$.

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The area bounded by the curve $y = {(x + 1)^2},\,y = {(x - 1)^2}$ and the line $y = \frac{1}{4}$ is

Let the eccentricity of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ be $\frac{5}{4}$. If the equation of the normal at the point $\left(\frac{8}{\sqrt{5}}, \frac{12}{5}\right)$ on the hyperbola is $8 \sqrt{5} x +\beta y =\lambda$, then $\lambda-\beta$ is equal to

Let $(a, b)$ be the point of intersection of the curve $x^{2}=2 y$ and the straight line $y-2 x-6=0$ in the second quadrant. Then the integral $I=\int_{a}^{b} \frac{9 x^{2}}{1+5^{x}} d x$ is equal to :

Let $p$ and $q$ be the position vectors of $P$ and $Q$ respectively with respect to $O$ and $|p|\, = p,\,\,|q|\,\, = q.$ The points $R$ and $S$ divide $PQ$ internally and externally in the ratio $2 : 3$ respectively. If $\overrightarrow {OR} $ and $\overrightarrow {OS} $ are perpendicular, then

The three points $(-2,2), \,(8,-2)$ and $(-4, -3)$ are the vertices of

The value of ${(1 + i)^8} + {(1 - i)^8}$ is

If the coordinates of the points $A,\, B, \,C$, be $(4,4), \,(3,-2)$ and $(3,-16)$ respectively, then the area of the triangle ABC is

The equation of the directrix of the parabola ${x^2} + 8y - 2x = 7$ is

If for all real values of $x,\,\,\frac{{4{x^2} + 1}}{{64{x^2} - 96x\,\sin \alpha + 5}}$ $ < \frac{1}{{32}},$ then $\alpha$ lies in the interval

If $x, y, z \in R^+$ are such that $z > y > x > 1$ , ${\log _y}x + {\log _x}y = \frac{5}{2}$ and ${\log _z}y + {\log _y}z = \frac{{10}}{3}$ then ${\log _x}z$ is equal to