Evaluate the following intregals: (1- )^3(2+ ) dx - Vidyadip

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 "5 Marks Questions" from INTEGRALS→INTEGRALS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side.

Differentiate the following functions with respect to x:
$\text{e}^{\sin\text{x}}+(\tan\text{x})^\text{x}$

Find the general solution of the differential equation $(1+\text{y}^2)+(\text{x}-\text{e}^{{\tan^{-1}\text{y}}})\frac{\text{dy}}{\text{dx}}=0.$

Show that $\text{y}=\text{A}\cos\text{x}+\text{B}\sin\text{x}$ is a solution of the differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{y}=0$

A manufacturer of patent medicines is preparing a production plan on medicines, $A$ and $B.$ There are sufficient raw materials available to make $20000$ bottles of $A$ and $40000$ bottles of $B,$ but there are only $45000$ bottles into which either of the medicines can be put. Further, it takes $3$ hours to prepare enough material to fill $1000$ bottles of $A,$ it takes $1$ hour to prepare enough material to fill $1000$ bottles of $B$ and there are $66$ hours available for this operation. The profit is $Rs. 8$ per bottle for $A$ and $Rs. 7$ per bottle for $B.$ How should the manufacturer schedule his production in order to maximize his profit?

Let R = {(a, a), (b, b), (c, c), (a, b)} be a relation on set A = a, b, c. Then, R is:

Identify relation.
Reflexive.
Symmetric.
Antisymmetric.

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}} = \tan(\text{x}+\text{y})$

In set $I \times I _0$, relation $R$ is defined such that $(a, b)$ $R (c, d) \Leftrightarrow a d=b c$ if $I _{ 0 }$ is set of non-zero integers. Then prove that $R$ is equivalence relation.

Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\frac{2\text{t}}{1+\text{t}^2}\text{ and y}=\frac{1-\text{t}^2}{1+\text{t}^2}$

A letter is known to have come either from $\text{LONDON}$ or $\text{CLIFTON}$. On the envelope just two consecutive letters $\text{ON}$ are visible. What is the probability that the letter has come from$, \text{LONDON}$.