Question
Evaluate the following integrals:
$\int\frac{1}{\sin\text{x}\cos^3\text{x}}\text{ dx}$
$\int\frac{1}{\sin\text{x}\cos^3\text{x}}\text{ dx}$
✓
Answer
$\frac{1}{\sin\text{x}\cos^3\text{x}}=\frac{\sin^2\text{x}+\cos^2\text{x}}{\sin\text{x}\cos^3\text{x}}$
$=\frac{\sin\text{x}}{\cos^3\text{x}}+\frac{1}{\sin\text{x}\cos\text{x}}$
$=\tan\text{x}\sec^2\text{x}+\frac{\frac{1\cos^2\text{x}}{\sin\text{x}\cos\text{x}}}{\cos^2\text{x}}$
$=\tan\text{x}\sec^2\text{x}+\frac{\sec^2\text{x}}{\tan\text{x}}$
$\therefore\ \int\frac{1}{\sin\text{x}\cos^3\text{x}}\text{ dx}=\int\tan\text{x}\sec^2\text{x}\text{ dx}+\int\frac{\sec^2\text{x}}{\tan\text{x}}\text{ dx}$
Let $\tan\text{x}=\text{t}$
$\sec^2\text{x}\text{ dx}=\text{dt}$
$\int\frac{1}{\sin\text{x}\cos^3\text{x}}\text{ dx}=\int\text{t}\text{ dt}+\int\frac{1}{\text{t}}\text{ dt}$
$=\frac{\text{t}^2}{2}+\log|\text{t}|+\text{C}$
$=\frac{1}{2}\tan^2\text{x}+\log|\tan\text{x}|+\text{C}$
$=\frac{\sin\text{x}}{\cos^3\text{x}}+\frac{1}{\sin\text{x}\cos\text{x}}$
$=\tan\text{x}\sec^2\text{x}+\frac{\frac{1\cos^2\text{x}}{\sin\text{x}\cos\text{x}}}{\cos^2\text{x}}$
$=\tan\text{x}\sec^2\text{x}+\frac{\sec^2\text{x}}{\tan\text{x}}$
$\therefore\ \int\frac{1}{\sin\text{x}\cos^3\text{x}}\text{ dx}=\int\tan\text{x}\sec^2\text{x}\text{ dx}+\int\frac{\sec^2\text{x}}{\tan\text{x}}\text{ dx}$
Let $\tan\text{x}=\text{t}$
$\sec^2\text{x}\text{ dx}=\text{dt}$
$\int\frac{1}{\sin\text{x}\cos^3\text{x}}\text{ dx}=\int\text{t}\text{ dt}+\int\frac{1}{\text{t}}\text{ dt}$
$=\frac{\text{t}^2}{2}+\log|\text{t}|+\text{C}$
$=\frac{1}{2}\tan^2\text{x}+\log|\tan\text{x}|+\text{C}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If $\text{x}=\text{a}\sin2\text{t}(1+\cos 2\text{t})$ and $\text{y}=\text{b}\cos\text{t}(1-\cos2\text{t}),$ show that at $\text{t}=\frac{\pi}{4},\frac{\text{dy}}{\text{dx}}=\frac{\text{b}}{\text{a}}\text{ t}=\frac{\pi}{4},\frac{\text{dy}}{\text{dx}}=\frac{\text{b}}{\text{a}}$
→Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\frac{1-\text{t}^2}{1+\text{t}^2}\text{ and y}=\frac{2\text{t}}{1+\text{t}^2}$
→$\text{x}=\frac{1-\text{t}^2}{1+\text{t}^2}\text{ and y}=\frac{2\text{t}}{1+\text{t}^2}$
Tow godowns$, A$ and $B,$ have grain storage capacity of $100$ quintals and $50$ quintals respectively. They supply to $3$ ration shops$, D, E$ and $F,$ whose requirements are $60, 50$ and $40$ quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:
How should the supplies be transported in order that the transportation cost is minimum?
→How should the supplies be transported in order that the transportation cost is minimum?
Evaluate : $\int_{-\pi}^\pi \frac{2 x(1+\sin x)}{1+\cos ^2 x}\ d x$
→Write the minimum value of $\text{f}(\text{x})=\text{x}^\frac{1}{\text{x}}.$
→A firm manufactures $3$ products $A, B$ and $C.$ The profits are $Rs. 3, Rs. 2$ and $Rs. 4$ respectively. The firm has $2$ machines and below is the required processing time in minutes for each machine on each product:
Machines $M_1$ and $M_{2 }$ have $2000$ and $2500$ machine minutes respectively. The firm must manufacture $100 A\ 's, 200 B\ 's$ and $50 C\ 's$ but not more than $150 A\ 's.$ Set up a $\text{LPP}$ to maximize the profit.
→|
Machine
|
Products
|
||
| $A$ | $B$ | $C$ | |
| $M_1$ | $4$ | $3$ | $5$ |
| $M_2$ | $2$ | $2$ | $4$ |
Evaluate the following integrals:
$\int_{0}^\limits{{\pi}}\frac{1}{5+3\cos\text{x}}\text{ dx}$
→$\int_{0}^\limits{{\pi}}\frac{1}{5+3\cos\text{x}}\text{ dx}$
Differentiate the following functions with respect to x:
$\sin^{-1}\big\{\sqrt{1-\text{x}^2}\big\},0<\text{x}<1$
→$\sin^{-1}\big\{\sqrt{1-\text{x}^2}\big\},0<\text{x}<1$
Evaluate the following integrals:
$\int^\limits{\frac{\pi}{2}}_{0}\frac{\cos\text{x}}{\big(\cos\frac{\text{x}}{2}+\sin\frac{\text{x}}{2}\big)^\text{n}}\text{ dx}$
→$\int^\limits{\frac{\pi}{2}}_{0}\frac{\cos\text{x}}{\big(\cos\frac{\text{x}}{2}+\sin\frac{\text{x}}{2}\big)^\text{n}}\text{ dx}$
Construct $a_{2 \times 2}$ matrix, where,
→- $\text{a}_\text{ij}=\frac{(\text{i}-2\text{j})^2}{2}$
- $\text{a}_\text{ij}=|-2\text{i}+3\text{j}|$