Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
Question
Evaluate the following integrals:
$\int\frac{\cot\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
✓

Answer

Let I $=\int\frac{\cot\text{x}}{\sqrt{\sin\text{x}}}\text{dx}\ .....(1)$
Let $\sin\text{x}=\text{t}$ then,
$\text{d}(\sin\text{x})=\text{dt}$
$\Rightarrow\cos\text{x}\text{ dx}=\text{dt}$
$\text{Now,}\text{I}=\int\frac{\cot\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
$=\int\frac{\cot\text{x}}{\sin\text{x}\sqrt{\sin\text{x}}}\text{dx}$
$\int\frac{\cos\text{x}}{(\sin\text{x})^{\frac{3}{2}}}\text{dx}$
$\Rightarrow\ =\int\frac{\cos\text{x}}{(\sin\text{x})^\frac{3}{2}}\text{dx}\ ...(2)$
Putting $\sin\text{x}=\text{t}$ and $\cos\text{x}\text{ dx}=\text{dt}$ in equation (2), we get
$\text{I}=\int\frac{\text{dt}}{\text{t}^\frac{3}{2}}$
$=\int\text{t}^{-\frac{3}{2}}\text{dt}$
$=-2\text{t}^{-\frac{1}{2}}+\text{C}$
$=\frac{-2}{\sqrt{\text{t}}}+\text{C}$
$=\frac{-2}{\sqrt{\sin\text{x}}}+\text{C}$
$\therefore\text{I}=\frac{-2}{\sqrt{\sin\text{x}}}+\text{C}$

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 "4 Marks" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

$\int\frac{\text{x}}{\sqrt{\text{x}+\text{a}}-\sqrt{\text{x}+\text{b}}}\text{dx}$
→
Integrate the function in Exercise:
$\text{e}^\text{x}\Bigg(\frac{1+\sin\text{x}}{1+\cos\text{x}}\Bigg)$
→
Prove that $\text{f(x)}=\begin{cases}\frac{\sin\text{x}}{\text{x}},&\text{x}<0\\\text{x}+1,&\text{x}\geq0\end{cases}$ is everywhere continuous.
→
Find the value of $\lambda$ so that the following vectors are coplanar:
$\vec{\text{a}}=\hat{\text{i}}+3\hat{\text{j}},\vec{\text{b}}=5\hat{\text{k}},\vec{\text{c}}=\lambda\hat{\text{i}}-\hat{\text{j}}$
→
If a young man drives his vehicle at 25 km/hr, he has to spend Rs. 2 per km on petrol. If he drives it at a faster speed of 40 km/hr, the petrol cost increases to Rs. 5 per km. He has Rs. 100 to spend on petrol and travel within one hour. Express this as an LPP and solve the same.
→
Find the distance of the point with position vector $-\hat{\text{i}}-5\hat{\text{j}}-10\hat{\text{k}}$ from the point of intersection of the line $\vec{\text{r}}=(2\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}})+\lambda(3\hat{\text{i}}+4\hat{\text{j}}+12\hat{\text{k}})$ with the plane $\vec{\text{r}}.(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}})=5.$
→
Form the differential equation of the family of circles touching the y-axis at origin.
→
Of all the closed cylindrical cans (right circular), which enclose a given volume of 100cm3, which has the minimum surface area ?
→
Show that the points whose position vectors are

$\vec{\text{a}}=4\hat{\text{i}}-3\hat{\text{j}}+\hat{\text{k}}, \vec{\text{b}}=2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}},\vec{\text{c}}=\hat{\text{i}}-\hat{\text{j}}$ from a right triangle.

→
Evaluate the following intregals:
$\int\frac{1}{\sin\text{x}-\cos\text{x}}\ \text{dx}$
→