Evaluate the following integrals: - 1+ 2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\cos\text{x}-\sin\text{x}}{1+\sin2\text{x}}\text{dx}$

✓

Answer

$\frac{\cos\text{x}-\sin\text{x}}{1+\sin2\text{x}}=\frac{\cos\text{x}-\sin\text{x}}{(\sin^2\text{x}+\cos^2\text{x})+2\sin\text{x}\cos\text{x}}$
$[\sin^2\text{x}+\cos^2\text{x}=1;\ \sin2\text{x}=2\sin\text{x}\cos\text{x}]$
$=\frac{\cos\text{x}-\sin\text{x}}{(\sin\text{x}+\cos\text{x})^2}$
$\sin\text{x}+\cos\text{x}=\text{t}$
$\therefore(\sin\text{x}+\cos\text{x})\text{dx}=\text{dt}$
$\Rightarrow\int\frac{\cos\text{x}-\sin\text{x}}{1+\sin2\text{x}}\text{dx}=\int\frac{\cos\text{x}-\sin\text{x}}{(\sin\text{x}+\cos\text{x})^2}\text{dx}$
$=\int\frac{\text{dt}}{\text{t}^2} $
$=\int\text{t}^{-2}\text{dt}$
$=-\text{t}^{-1}+\text{C}$
$=-\frac{1}{\text{t}}+\text{C}$
$=\frac{-1}{\sin\text{x}+\cos\text{x}}+\text{C}$

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.(3 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

In the following, determine the values of constants involved in the definition so that the given function is continuous:
$\text{f(x)}=\begin{cases}\frac{\sin2\text{x}}{5\text{x}},&\text{if }\text{ x}\neq0\\3\text{k},&\text{if }\text{ x}=0\end{cases}$

Find the equation of the line passing through the points (2, 1, 3) and perpendicular to the lines $\frac{\text{x}-1}{1}=\frac{\text{y}-2}{2}=\frac{\text{z}-3}{3}$ and $\frac{\text{x}}{-3}=\frac{\text{y}}{2}=\frac{\text{z}}{5}$

The position vectors $\mathrm{f}$ three consecutive vertices of a parallelogram are $\hat{i}+\hat{j}+\hat{k}$

$\hat{i}+3 \hat{j}+5 \hat{k}$ and $7 \hat{i}+9 \hat{j}+11 \hat{k}$ Find the position vector of the fourth vertex.

Differentiate the following functions with respect to x:
$\tan^{-1}(\text{e}^{\text{x}})$

Three coins are tossed simultaneously, $X$ is the number of heads. Find expected value and variance of $X$.

If $e^{x+y} - x = 0$, prove that $\frac{\text{dy}}{\text{dx}}=\frac{1-\text{x}}{\text{x}}$

If $\text{A}=\begin{bmatrix}\cos\text{x}&\sin\text{x}\\-\sin\text{x}&\cos\text{x}\end{bmatrix},$ find x satisfying $0<\text{x}<\frac{\pi}{2}$ when $A + A^T = I$

Write a value of $\int\tan^6\text{x}\sec^2\text{x}\text{ dx}$

$\int_0^{\pi / 2} \frac{\cos X}{(1+\sin x)(2+\sin x)} \cdot d x$

For each of the following differential equations, find the particular solution satisfying the given condition:

$\left(x-y^2 x\right) d x-\left(y+x^2 y\right) d y=0$, when $x=2, y=0$