Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS4 Marks
Question
Evaluate the following intregals:
$\int\frac{\cos\text{x}}{(1-\sin\text{x})^3(2+\sin\text{x})}\ \text{dx}$
✓

Answer

Let
$\sin\text{x}=\text{t}$
$\Rightarrow\cos\text{x}=\text{dt}$
$\therefore\int\frac{\cos\text{x}}{(1+\sin\text{x})^3(2+\sin\text{x})}=\int\frac{1}{(1-\text{t})^3(2+\text{t})}\ \text{dt}$
Let $\text{f}(\text{t})=\frac{1}{(1-\text{t})^3(2+\text{t})}$
Then suppose
$\frac{1}{(1-\text{t})^3(2+\text{t})}=\frac{\text{A}}{1-\text{t}}+\frac{\text{B}}{(1-\text{t})^2}+\frac{\text{C}}{(1-\text{t})^3}+\frac{\text{D}}{(2+\text{t})}$
$\Rightarrow1=\text{A}(1-\text{t})^2(2+\text{t})+\text{B}(1-\text{t})(2+\text{t})\\+\text{C}(2+\text{t})+\text{D}(1-\text{t})^3$
Put t = 1
1 = 27D
$\Rightarrow\text{D}=\frac{1}{27}$
Similarly, we can find that $\text{A}=\frac{-1}{27}$ and $\text{B}=\frac{+1}{9}$
$\therefore\int\frac{1}{(1-\text{t})^3(2+\text{t})}\ \text{dt}=\frac{-1}{27}\int\frac{1}{1-\text{t}}\ \text{dt}+\frac{1}{9}\int\frac{\text{dt}}{(1-\text{t})^2}\\+\frac{1}{3}\int\frac{\text{dt}}{(1-\text{t})^3}+\frac{1}{27}\int\frac{\text{dt}}{2+\text{t}}$
$=\frac{-1}{27}\log|1-\text{t}|+\frac{1}{9(1-\text{t})}+\frac{1}{6(1-\text{t})^2}+\frac{1}{27}\log|2+\text{t}|+\text{C}$
Putting $\text{t}=\sin\text{x}$ we get
$\int\frac{\cos\text{x}}{(1-\sin\text{x})^3(2+\sin\text{x})}\ \text{dx}$
$=\frac{-1}{27}\log|1-\sin\text{x}|+\frac{1}{9(1-\sin\text{x})}\\+\frac{1}{6(1-\sin\text{x})^2}+\frac{1}{27}\log|2+\sin\text{x}|+\text{C}$

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 INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the equation of the plane that bisects the line segment joining the points (1, 2, 3) and (3, 4, 5) and is at right angle to it.
→
Find the intervals in which the following functions are increasing or decreasing.
$\text{f}(\text{x})=5\text{x}^{\frac{3}{2}}-3\text{x}^{\frac{5}{2}},\text{x}>0$
→
Write the vector equation of the following lines and hence determine the distance between them $\frac{\text{x}-1}{2}=\frac{\text{y}-2}{3}=\frac{\text{z}+4}{6}$ and $\frac{\text{x}-3}{4}=\frac{\text{y}-3}{6}=\frac{\text{z}+5}{12}$
→
A firm has to transport at least 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost of engaging each large van is Rs 400 and each small van is Rs 200. Not more than Rs 3000 is to be spent daily on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimize cost.
→
Find the equation of the plane which bisects the line segment joining the points (-1, 2, 3) and (3, -5, 6) at right angles.
→
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
→
Find the area enclosed by the curve y = |x - 1| and y = -|x - 1 | + 1.
→
Maximum Z = 2x + 4y

Subject to

$\text{x}+\text{y}\geq8$

$\text{x}+4\text{y}\geq12$

$\text{x}\geq3,\text{y}\geq2$

→
Find the angle between the lines whose direction cosines are given by the equations:
l + m +n = 0 and l2 + m2 + n2 = 0
→
Evaluate the following integrals:
$\int\limits_{0}^{\text{a}}\frac{\text{x}}{\sqrt{\text{a}^2+\text{x}^2}}\text{ dx}$
→