_ ^ d ^3 = - Vidyadip

MCQ

$\int_{}^{} {\frac{{d\theta }}{{\sin \theta {{\cos }^3}\theta }} = } $

A
$\log \tan \theta + {\tan ^2}\theta + c$
B
$\log \tan \theta - \frac{1}{2}{\tan ^2}\theta + c$
✓
$\log \tan \theta + \frac{1}{2}{\tan ^2}\theta + c$
D
None of these

✓

Answer

Correct option: C.

$\log \tan \theta + \frac{1}{2}{\tan ^2}\theta + c$

c
(c)$\int_{}^{} {\frac{{d\theta }}{{\sin \theta {{\cos }^3}\theta }} = \int_{}^{} {\frac{{{{\sec }^2}\theta \,d\theta }}{{\sin \theta \cos \theta }} = \int_{}^{} {\frac{{{{\sec }^2}\theta (1 + {{\tan }^2}\theta )}}{{\tan \theta }}} } } {\rm{ }}d\theta $
Put $t = \tan \theta \Rightarrow dt = {\sec ^2}\theta \,d\theta ,$ then it reduces to
$\int_{}^{} {\frac{{1 + {t^2}}}{t}\,dt = \int_{}^{} {\left( {\frac{1}{t} + t} \right)\,dt} } $
$ = \log t + \frac{{{t^2}}}{2} + c = \log \tan \theta + \frac{{{{\tan }^2}\theta }}{2} + 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

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

If the system of equations $x+y+z=6 \,; \,2 x+5 y+\alpha z=\beta \,; \, x+2 y+3 z=14$ has infinitely many solutions, then $\alpha+\beta$ is equal to.

Which of the following six statements are true about the cubic polynomial

$P(x) = 2x^3 + x^2 + 3x - 2? $

$(i)$ It has exactly one positive real root.

$(ii)$ It has either one or three negative roots.

$(iii)$It has a root between $0$ and $1.$

$(iv)$ It must have exactly two real roots.

$(v)$ It has a negative root between $- 2$ and $-1.$

$(vi)$ It has no complex roots.

The vector equation of $X Y$-plane is

$\int_{}^{} {\frac{{3\cos x + 3\sin x}}{{4\sin x + 5\cos x}}\;dx = } $

If $\frac{{{d^2}y}}{{d{x^2}}} = 0,$ then

If A and B are two events such that $\text{P(A)}=\frac{1}{2},\text{P(B)}=\frac{1}{3},\text{P}(\text{A}|\text{B})=\frac{1}{4},$ then $\text{P}(\overline{\text{A}}\cap\overline{\text{B}})$ equals.

$\frac{1}{12}$
$\frac{3}{4}$
$\frac{1}{4}$
$\frac{3}{16}$

The area bounded by curves $y = \cos x$ and $y = \sin x$ and ordinates $x = 0$ and $x = \frac{\pi }{4}$ is

If $A = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right],B = \left[ {\begin{array}{*{20}{c}}0&{ - i}\\i&0\end{array}} \right]$ then ${(A + B)^2}$ equals

The vectors $2\,i + 3\,j - 4\,k$ and $a\,i + b\,j + c\,k$ are perpendicular, when

Let $S$ be the region bounded by the curves $y=x^{3}$ and $y ^{2}= x$. The curve $y =2| x |$ divides $S$ into two regions of areas $R_{1}$ and $R_{2}$.

If $\max \left\{R_{1}, R_{2}\right\}=R_{2}$, then $\frac{R_{2}}{R_{1}}$ is equal to