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 "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

$\int\frac{3\text{x}+5}{\sqrt{7\text{x}+9}}\text{dx}$

If $\text{x}=\cot\text{t and y}=\sin\text{t},$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{1}{\sqrt{3}}\text{ at t}=\frac{2\pi}{3}$

Find the area lying above the x-axis and under the paraola $y^2 = 4x - x^2$.

Solve the matrix equations:
$\begin{bmatrix}1&2&1\end{bmatrix}\begin{bmatrix}1&2&0\\2&0&1\\1&0&2\end{bmatrix}\begin{bmatrix}0\\2\\\text{x}\end{bmatrix}=0$

Prove that:
$\begin{vmatrix}\text{b}+\text{c}&\text{a}-\text{b}&\text{a}\\\text{c}+\text{a}&\text{b}-\text{c}&\text{b}\\\text{a}+\text{b}&\text{c}-\text{a}&\text{c}\end{vmatrix}=3\text{abc}-\text{a}^3-\text{b}^3-\text{c}^3$

Given the probability that A can solve a problem is $\frac{2}{3}$ and the probability that B can solve the same problem is $\frac{3}{5}$. Find the probability that none of the two will be able to solve the problem.

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

Find the absolute maximum and the absolute minimum value of the following functions in the given intervals:
$\text{f}(\text{x})=4\text{x}-\frac{\text{x}^{2}}{2}\ \text{in}\ [2,4,5]$

Let $\text{A}=\begin{bmatrix}1&1&1\\0&1&1\\0&0&1\end{bmatrix}$ Use the principle of mathematical induction to show that
$\text{A}^\text{n}=\begin{bmatrix}1&\text{n}&\frac{\text{n}(\text{n}+1)}{2}\\0&1&\text{n}\\0&0&1\end{bmatrix}$ for every positive integer n.

Evaluate the following integrals:$\int\limits^{\pi}_0\text{x}\sin\text{x}\cos^2\text{x dx}$