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

Question

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

✓

Answer

We have,
$\text{I}=\int\frac{\cos\text{x}\text{ dx}}{(1-\sin\text{x})(2-\sin\text{x})}$
putting $\sin\text{x}=\text{t}$
$\Rightarrow\cos\text{x dx}=\text{dt}$
$\therefore\text{I}=\int\frac{\text{dt}}{(1-\text{t})(2-\text{t})}$
$\therefore\text{I}=\int\frac{\text{dt}}{(1-\text{t})(2-\text{t})}$
Let $\frac{1}{(\text{t}-1)(\text{t}-2)}=\frac{\text{A}}{\text{t}-1}+\frac{ \text{B}}{\text{t}-2}$
$\Rightarrow\frac{1}{(\text{t}-1)(\text{t}-2)}=\frac{\text{A}(\text{t}-2)+\text{B}(\text{t}-1)}{(\text{t}-1)(\text{t}-2)}$
$\Rightarrow1=\text{A}(\text{t}-2)+\text{B}(\text{t}-1)$
Putting t - 1 = 0
⇒ t = 1
$\therefore$ 1 = A (1 - 2) + B × 0
⇒ A = -1
Putting t - 2 = 0
⇒ t = 2
$\therefore$ 1 = A × 0 + B(2 - 1)
⇒ B = 1
$\therefore\text{I}=\int\frac{-\text{dt}}{\text{t}-1}+\int\frac{\text{dt}}{\text{t}-2}$
$=-\log|\text{t}-1|+\log|\text{t}-2|+\text{C}$
$=\log\Big|\frac{\text{t}-2}{\text{t}-1}\Big|+\text{C}$
$=\log\Big|\frac{\sin\text{x}-2}{\sin\text{x}-1}\Big|+\text{C}$
$=\log\Big|\frac{2-\sin\text{x}}{1-\sin\text{x}}\Big|+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate : $\int \frac{1}{2-3 \sin 2 x} \cdot d x$

If $\text{x}=\sin\text{t},\text{y}=\sin\text{pt}$ prove that $(1-\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{x}\frac{\text{dy}}{\text{dx}}+\text{p}^2\text{y}=0$

Form the differential equation corresponding to $\text{y}=\text{e}^{\text{mx}}$ by eliminating m.

Evaluate the following integrals:
$\int\frac{\cos^3\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$

If $\vec{\text{a}}$ be the position vector whose tip is (5, -3), find the coordinates of a point B such that $\overrightarrow{\text{AB}}=\vec{\text{a}}$, the coordinates of A being (4, -1).

In $\triangle ABC$ prove that $\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}=\frac{a+b+c}{b+c-a} \cot \frac{A}{2}$

Find the area of the region included between:

$y=x^2+3$ and $y=x+3$

Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,
$\text{f}(\text{x})=\text{x}\sqrt{2-\text{x}^{2}}-\sqrt{2}\leq\text{x}\leq\sqrt{2}$

For the following differntial equations verify that the accompanying function is a solution:

Differential equation	Function
$\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\text{y}^2$	$\text{y}=\frac{\text{a}}{\text{x}+\text{a}}$

Differentiate the following functions with respect to x:
$\sin^{-1}\Big(\frac{\text{x}+\sqrt{1-\text{x}^3}}{\sqrt{2}}\Big),-1<\text{x}<1$