Download App

Indefinite Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIndefinite Integrals5 Marks
Question
Evaluvate the following intregals:
$\int\frac{3+2\cos\text{x}+4\sin\text{x}}{2\sin\text{x}+\cos\text{x}+3}\ \text{dx}$
✓

Answer

Let $\text{I}=\int\frac{3+2\cos\text{x}+4\sin\text{x}}{2\sin\text{x}+\cos\text{x}+3}\ \text{dx}$
Let $3+2\cos\text{x}+4\sin\text{x}=\lambda\frac{\text{d}}{\text{dx}}(2\sin\text{x}+\cos\text{x}+3)+\mu(2\sin\text{x}+\cos\text{x}+3)+\text{v}$
$3+2\cos\text{x}+4\sin\text{x}=\lambda(2\cos\text{x}-\sin\text{x})+\mu(2\sin\text{x}+\cos\text{x}+3)+\text{v}$
$3+2\cos\text{x}+4\sin\text{x}=(-\lambda+2\mu)\sin\text{x}+(2\lambda+\mu)\cos\text{x}+3\mu+\text{v}$
Comparing the coefficient of $\sin\text{x}\ \&\cos\text{x}$ on the both the sides,
$-\lambda+2\mu\ =4\dots\dots(1)$
$2\lambda + \mu = 2 \ \dots\dots(2)$
$2\mu + \text{v} = 3 \ \dots\dots(3)$
Solving Equation (1), (2) and (3), we get
$\lambda=0,\mu=2,\text{v}=-3$
$\text{I}==\int\frac{2(2\sin\text{x}+\cos\text{x}+3)-3}{(2\sin\text{x}+\cos\text{x}+3)}\ \text{dx}$
$=2\int\text{dx}-2\int\frac{1}{2\sin\text{x}+\cos\text{x}+3}\ \text{dx}$
$\text{I}=2\text{x}-3\text{I}_1+\text{C}_1\dots\dots(4)$
Let $\text{I}_1=\int\frac{1}{2\sin\text{x}+\cos\text{x}+3}\ \text{dx}$
$\sin\text{x}=\frac{2\tan\frac{\text{x}}{2}}{\Big(1+\tan^2\frac{\text{x}}{2}\Big)},\cos\text{x}=\frac{1-\tan^2\frac{\text{x}}{2}}{\Big(1+\tan^2\frac{\text{x}}{2}\Big)}$
$\text{I}_1=\int\frac{1}{2\Bigg(\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\Bigg)+\Bigg(\frac{1-\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\Bigg)+3}\ \text{dx}$
$=\int\frac{\Big(1+\tan^2\frac{\text{x}}{2}\Big)}{4\tan\frac{\text{x}}{2}+1-\tan^2\frac{\text{x}}{2}+3\Big(1+\tan^2\frac{\text{x}}{2}\Big)}\ \text{dx}$
$=\int\frac{\sec^2\frac{\text{x}}{2}}{2\tan^2\frac{\text{x}}{2}+4\tan\frac{\text{x}}{2}+4}\ \text{dx}$
Let $\tan\frac{\text{x}}{2}=\text{t}$
$\frac{1}{2}\sec^2\frac{\text{x}}{2}=\text{dt}$
$\text{I}_1=\int\frac{2\text{dt}}{2\text{t}^2+4\text{t}+4}$
$\frac{2}{2}\int\frac{\text{dt}}{\text{t}^2+2\text{t}+2}$
$=\int\frac{\text{dt}}{\text{t}^2+2\text{t}+1-1+2}$
$=\int\frac{\text{dt}}{(\text{t}+1)^2+1}$
$=\tan^{-1}\Big(\tan\frac{\text{x}}{2}+1)+\text{C}_2$
Now, using equation (1),
$\text{I}=2\times-3\tan^{-1}\Big(\tan\frac{\text{x}}{2}+1\Big)+\text{C}$

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 Indefinite Integrals→Indefinite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

An aeroplane can carry a maximum of 200 passengers. A profit of Rs.1000 is made on each executive class ticket and a profit of Rs.600 is made on each economy class ticket. The airline reserves atleast 20 seats for executive class. However, atleast 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit of the airline. What is the maximum profit?
→
If a young man drives his vehicle at 25 km/hr, he has to spend Rs. 2 per km on petrol. If he drives it at a faster speed of 40 km/hr, the petrol cost increases to Rs. 5 per km. He has Rs. 100 to spend on petrol and travel within one hour. Express this as an LPP and solve the same.
→
Show that the derivative of the function $f$ given by $f(x) = 2x^3 - 9x^2 + 12x + 9,$ at $x = 1$ and $x = 2$ are equal.
→
For the matrix $\text{A}=\begin{bmatrix}1 & -1 & 1 \\2 & 3 & 0 \\ 18 & 2 & 10 \end{bmatrix},$ show that A (adjoint A) = 0.
→
Solve the following system of equations by matrix method:$\frac{2}{\text{x}}-\frac{3}{\text{y}}+\frac{3}{\text{z}}=10$
$\frac{1}{\text{x}}+\frac{1}{\text{y}}+\frac{1}{\text{z}}=10$
$\frac{3}{\text{x}}-\frac{1}{\text{y}}+\frac{2}{\text{z}}=13$
→
Differentiate $\sin^{-1}\Big(4\text{x}\sqrt{1-4\text{x}^2}\Big)$ with respect to $\sqrt{1-4\text{x}^2},$ if:
$\text{x}\in\Big(-\frac{1}{2},-\frac{1}{2\sqrt{2}}\Big)$
→
If $\text{f(x)}=\frac{\tan\big(\frac{\pi}{4}-\text{x}\big)}{\cot2\text{x}}$ for $\text{x}\neq\frac{\pi}{4},$ find the value which can be assigned to f(x) at $\text{x}=\frac{\pi}{4}$ so that the function f(x) becomes continuous every where in $\Big[0,\frac{\pi}{2}\Big]$  
→
If $\text{x}=\cos\text{t}+\log\tan\Big(\frac{\text{t}}{2}\Big),\ \text{y}=\sin\text{t},$ then find the values of $\frac{\text{d}^2\text{y}}{\text{dt}^2}$ and $\frac{\text{d}^2\text{y}}{\text{dx}^2}$ at $\text{t}=\frac{\pi}{4}.$
→
If $\text{A}=\begin{bmatrix}2&3\\1&2\end{bmatrix}$ and $\text{I}=\begin{bmatrix}1&0\\0&1\end{bmatrix},$ then find $\lambda,\mu$ so that $\text{A}^2=\lambda\text{A}+\mu\text{I}$
→
Find the equation of the plane through the line of intersection of the planes $\vec{\text{r}}\cdot(\hat{\text{i}}+3\hat{\text{j}})+6=0$ and $\vec{\text{r}}\cdot(3\hat{\text{i}}-\hat{\text{j}}-4\hat{\text{k}})=0,$ which is at a unit distance from the origin.
→