_ ^ dx ( x + 2x) = - Vidyadip

MCQ

$\int_{}^{} {\frac{{dx}}{{(\sin x + \sin 2x)}} = } $

✓
$\frac{1}{6}\log (1 - \cos x) + \frac{1}{2}\log (1 + \cos x) - \frac{2}{3}\log (1 + 2\cos x)$
B
$6\log (1 - \cos x) + 2\log (1 + \cos x) - \frac{2}{3}\log (1 + 2\cos x)$
C
$6\log (1 - \cos x) + \frac{1}{2}\log (1 + \cos x) + \frac{2}{3}\log (1 + 2\cos x)$
D
None of these

✓

Answer

Correct option: A.

$\frac{1}{6}\log (1 - \cos x) + \frac{1}{2}\log (1 + \cos x) - \frac{2}{3}\log (1 + 2\cos x)$

a
(a)$I = \int_{}^{} {\frac{{dx}}{{\sin x(1 + 2\cos x)}}} = \int_{}^{} {\frac{{\sin x\,dx}}{{{{\sin }^2}x(1 + 2\cos x)}}} $
$ = \int_{}^{} {\frac{{\sin x\,dx}}{{(1 - \cos x)(1 + \cos x)(1 + 2\cos x)}}} $
Now differential coefficient of $\cos x$ is $ - \sin x$ which is given in numerator and hence we make the substitution $\cos x = t \Rightarrow - \sin x\,dx = dt$
$\therefore \,\,\,I = - \int_{}^{} {\frac{{dt}}{{(1 - t)(1 + t)(1 + 2t)}}} $
We split the integrand into partial fractions
$I = - \int {\left[ {\frac{1}{{6(1 - t)}} - \frac{1}{{2(1 + t)}} + \frac{4}{{3(1 + 2t)}}} \right]} \,dt$ etc.

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 7.1 inderfinite integral→7.1 inderfinite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $\vec a\, = \,\hat i\, + \,\hat j\, + \,\sqrt 2 \hat k,\,\,\vec b\, = \,{b_1}\hat i\, + \,{b_2}\hat j\, + \sqrt 2 \hat k$ and $\vec c\, = \,5\hat i\, + \,\hat j + \sqrt 2 \hat k$ be three vectors such that the projection vector of $\vec b$ on $\vec a$ is $\vec a$. If $\vec a\, + \vec b$ is perpendicular to $\vec c$ , then $\left| {\vec b} \right|$ is equal to

Let $9$ distinct balls be distributed among $4$ boxes, $B_{1}, B_{2}, B_{3}$ and $B_{4}$. If the probability that $B_{3}$ contains exactly $3$ balls is $k\left(\frac{3}{4}\right)^{9}$ then $\mathrm{k}$ lies in the set:

The eqution of the plane through the line x + y + 3 = 0 = 2x - y + 3z + 1 and parallel to the line $\frac{\text{x}}{1}=\frac{\text{y}}{2}=\frac{\text{z}}{3}$ is:

x - 5y + 3z = 7
x - 5y + 3z = -7
x + 5y + 3z = 7
x + 5y + 3z = -7

If $f(x)$ is a differentiable function, then $\mathop {\lim }\limits_{x \to a} {{af(x) - xf(a)} \over {x - a}}$ is

If ${I_1} = \int\limits_1^{\sin \theta } {\frac{x}{{1 + x^2}}} \,dx$ and ${I_2} = \int\limits_1^{\cos ec\theta } {\frac{{dx}}{{x\left( {{x^2} + 1} \right)}}}$; then the value of $\left| {\begin{array}{*{20}{c}}
{{I_1}}&{I_1^2}&{{I_2}} \\
{{e^{{I_1} + {I_2}}}}&{I_2^2}&{ - 1} \\
1&{I_1^2 + I_2^2}&{ - 1}
\end{array}} \right|$ is

The points A(1, 1, 0), B(0, 1, 1), C(1, 0, 1) and $\text{D}\big(\frac{2}{3},\frac{2}{3},\frac{2}{3}\big)$

Coplanar
Non-coplanar
Vertices of a parallelogram
None of these

The integral $\int \limits_{1}^{2} e ^{ x } \cdot x ^{ x }\left(2+\log _{ e } x \right) d x$ equal

Let the area enclosed by the lines $x + y =2, y =0$, $x=0$ and the curve $f(x)=\min \left\{x^2+\frac{3}{4}, 1+[x]\right\}$ where $[ x ]$ denotes the greatest integer $\leq x$, be $A$. Then the value of $12\,A$ is $............$.

Which of the following functions is decreasing on $\Big(0,\frac{\pi}{2}\Big)?$

Choose the correct answer from the given four options.
A die is thrown and a card is selected at random from a deck of 52 playing cards. The probability of getting an even number on the die and a spade card is: