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

Question

Evaluate the following intregals:
$\int\frac{\sin2\text{x}}{(1+\sin\text{x})(2+\sin\text{x})}\text{ dx}$

✓

Answer

Let $\int\frac{\sin2\text{x}}{(1+\sin\text{x})(2+\sin\text{x})}\text{ dx}=\frac{\text{A}}{1+\sin\text{x}}+\frac{\text{B}}{2+\sin\text{x}}$
$\Rightarrow\sin2\text{x}=\text{A}(2+\sin\text{B})+\text{B}(1+\sin\text{B})$
$\Rightarrow2\sin\text{x}\cos\text{x}=(2\text{A}+\text{B})+(\text{A}+\text{B})\sin\text{x}$
Equating similar terms, we get,
$2\text{A}+\text{B}=0\Rightarrow\text{B}=-2\text{A}\text{ and}$
$\text{A}+\text{B}=2\cos\Rightarrow-\text{A}=2\cos\text{x}$
$\Rightarrow\text{A}=-2\cos\text{x}$
Thus,
$\text{I}=\int-\frac{2\cos\text{x}}{1+\sin\text{x}}\text{ dx}+\int\frac{4\cos\text{x}}{1+\sin\text{x}}\text{ dx}$
$=-2\log|1+\sin\text{x}|+4\log|2+\sin\text{x}|+\text{C}$
$\text{I}=\log\Big|\frac{(2+\sin\text{x})^4}{(1+\sin\text{x})^2}\Big|+\text{C}$

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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Using integration find the area of the region bounded by the curves $\text{y} = \sqrt{4 - \text{x}^{2}}, \text{x}^{2} + \text{y}^{2} \text{4x} = 0$ and the x-axis.

Show that the function g(x) = x - [x] is discontinuous at all integral points. Here [x] denotes the greatest integer function.

Evalute the following integrals:
$\int\frac{1}{\cos(\text{x}+\text{a})\cos(\text{x}+\text{b})}\text{dx}$

Show that A′A and AA′ are both symmetric matrices for any matrix A.

Evaluvate the following intregals:
$\int\frac{2\tan\text{x}+3}{3\tan\text{x}+4}\ \text{dx}$

Two like parallel forces $\overrightarrow{a}$ and$\overrightarrow{b}$ act on a rigid body at A and B respectively. If $\overrightarrow{P}$ and $\overrightarrow{Q}$ are interchanged in position, show that the point of application of the resultant will be displaced through a distance $\frac{P -Q}{P + Q}.AB$

Solve the following determinant equations:
$\begin{vmatrix}3&-2&\sin(3\theta)\\-7&8&\cos(2\theta)\\-11&14&2\end{vmatrix}=0$

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman's time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman's time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman's time. If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find the number of tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as an LPP and solve it graphically.

If $\text{y}=\text{x}\sin\text{y},$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}(1-\text{x}\cos\text{y})}$

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}1&43&6\\7&35&4\\3&17&2\end{vmatrix}$