Evalute the following integrals: - +2 2 + dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{-\sin\text{x}+2\cos\text{x}}{2\sin\text{x}+\cos\text{x}}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{-\sin\text{x}+2\cos\text{x}}{2\sin\text{x}+\cos\text{x}}\text{dx}\ .....\text{(i)}$
Let $2\sin\text{x}+\cos\text{x}=\text{t}$ then,
$\text{d}(2\sin\text{x}+\cos\text{x})=\text{dt}$
$\Rightarrow(2\cos\text{x}-\sin\text{x})\text{dx}=\text{dt}$
$\Rightarrow\text{dx}=\frac{\text{dt}}{-\sin\text{x}+2\cos\text{x}}$
Putting $2\sin\text{x}+\cos\text{x}=\text{t and dx}=\frac{\text{dt}}{-\sin\text{x}+2\cos\text{x}}$ in equation (i), we get,
$\text{I}=\int\frac{-\sin\text{x}+2\cos\text{x}}{\text{t}}\times\frac{\text{dt}}{-\sin\text{x}+2\cos\text{x}}$
$=\int\frac{\text{dt}}{\text{t}}$
$=\log|\text{t}|+\text{C}$
$=\log|2\sin\text{x}+\cos\text{x}|+\text{C}$
$\therefore\text{I}=\log|2\sin\text{x}+\cos\text{x}|+\text{C}$

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

Two schools $A$ and $B$ want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school $A$ wants to award $₹ x$ each, $₹ y$ each and $₹ z$ each for the three respective values to $3, 2$ and $1$ students respectively with a total award money of $₹ 1,600$. School $B$ wants to spend $₹ 2,300$ to award its $4, 1$ and $3$ students on the respective values $($by giving the same award money to the three values as before$).$ If the total amount of award for one prize on each value is $₹ 900,$ using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.

If O is the origin and the coordinates of A are (a, b, c) Find the direction cosines of OA and the equation of the plane through A at right angles to OA.

Differentiate the function $(\sin x)^{x} + \sin^{-1} \sqrt{x}$ with respect to x.

Find $\frac{\text{dy}}{\text{ dx}} $in the following:
$\text{y}=\sin^{-1}\big(2\text{x}\sqrt{1-\text{x}^{2}}\big), -\frac{1}{\sqrt{2}}<\text{x}<\frac{1}{\sqrt{2}}$

A manufacturing company makes two models $A$ and $B$ of a product. Each piece of model $A$ requires $9$ labour hours for fabricating and $1$ labour hour for finishing. Each piece of model B requires $12$ labour hours for fabricating and $3$ labour hours for finishing. For fabricating and finishing, the maximum labour hours available are $180$ and $30$ respectively. The company makes a profit of $Rs. 8000$ on each piece of model $A$ and $Rs. 12000$ on each piece of model $B.$ How many pieces of model $A$ and model $B$ should be manufactured per week to realise a maximum profit? What is the maximum profit per week?

Solve the following initial value problems:
$\Big\{\text{x}\sin^2\Big(\frac{\text{y}}{\text{x}}\Big)-\text{y}\Big\}\text{dx + x dy}=0,\text{y}(1)=\frac{\pi}4$

Evaluate $\int\limits_{0}^{2}\text{(x}^{2}+\text{x}+2)\text{dx}$ as limit of sums.

Without expanding, show that the values of the following determinant are zero: $\begin{vmatrix}1^2&2^2&3^2&4^2\\2^2&3^2&4^2&5^2\\3^2&4^2&5^2&6^2\\4^2&5^2&6^2&7^2\end{vmatrix}$

Let $\vec{\text{a}}=\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}},\vec{\text{b}}=3\hat{\text{i}}-2\hat{\text{j}}+7\hat{\text{k}}$ and $\vec{\text{c}}=2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}.$Find a vector $\vec{\text{d}}$ which is perpendicular to both $\vec{\text{a}}$ and $\vec{\text{d}}$ and $\vec{\text{c}}.\vec{\text{d}}=15.$

Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix}2 & 5 \\ 1 & 3 \end{bmatrix}$