Evaluate the following integrals: - 8- 2x dx - Vidyadip

Question

Evaluate the following integrals:$\int\frac{\cos\text{x}-\sin\text{x}}{\sqrt{8-\sin2\text{x}}}\text{ dx}$

✓

Answer

Let $\text{I}=\int\frac{\cos\text{x}-\sin\text{x}}{\sqrt{8-\sin2\text{x}}}\text{ dx}$
$=\int\frac{\cos\text{x}-\sin\text{x}}{\sqrt{9-1-\sin2\text{x}}}\text{ dx}$
$=\int\frac{\cos\text{x}-\sin\text{x}}{\sqrt{9-\sin^2\text{x}-\cos^2\text{x}-2\sin\text{x}\cos\text{x}}}\text{ dx}$
$=\int\frac{\cos\text{x}-\sin\text{x}}{\sqrt{9-(\sin\text{x}+\cos\text{x})^2}}\text{ dx}$
Let $(\sin\text{x}+\cos\text{x})=\text{t}$
On differentiating both sides, we get
$(\cos\text{x}-\sin\text{x})\text{dx}=\text{dt}$
$\therefore\ \text{I}=\int\frac{1}{(3)^2-(\text{t})^2}\text{ dt}$
$=\sin^{-1}\Big(\frac{\text{t}}{3}\Big)+\text{C}$
$=\sin^{-1}\Big(\frac{\sin\text{x}-\cos\text{x}}{3}\Big)+\text{C}$
Hence, $\int\frac{\cos\text{x}-\sin\text{x}}{\sqrt{8-\sin2\text{x}}}\text{ dx}=\sin^{-1}\Big(\frac{\cos\text{x}-\sin\text{x}}{3}\Big)+\text{C}$

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 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

A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?
A = the number of heads is two,
B = the last throw results in head.

$\text{If y} = \sin (\log x) , \text{prove that}$$x^{2} \frac{d^{2}y}{dx^{2}} + x \frac{dy}{dx} + y =0$

If $\tan^{-1}\bigg(\frac{\text{x} - 2}{\text{x} - 4}\bigg) + \tan^{-1}\bigg(\frac{\text{x} + 2 }{\text{x} + 4 }\bigg) = \frac{\pi}{4} , $find the value of x.

If $\text{A}=\begin{bmatrix}1&2\\-2&1\end{bmatrix},\ \text{B}=\begin{bmatrix}2&3\\3&-4\end{bmatrix}$ and $\text{C}=\begin{bmatrix}1&0\\-1&0\end{bmatrix},$ verify $(\text{AB})\text{C}=\text{A}(\text{BC}).$

Find the area, lying above $x-$axis and included between the circle $x^2 + y^2 = 8x$ and the parabola $y^2 = 4x.$

Find the area of the ragion bounded by $x^2 = 16y, y = 1, y = 4$ and the parabola $y-$axis and the first quadrant.

Differentiate the following functions with respect to x:
$\text{x}^{\text{x}\cos\text{x}}+\frac{\text{x}^2+1}{\text{x}^2-1}$

Differentiate the following functions with respect to x:
$\text{e}^{\text{ax}}\sec\text{x}\tan2\text{x}$

If $\text{y}=\sin(\sin\text{x})$ prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\tan\text{x}.\frac{\text{dy}}{\text{dx}}+\text{y}\cos^2\text{x}=0$

Examine the continuity of the function
$\text{f}\text{(x)}=\begin{cases}3\text{x}-2 &, \text{ if x} \leq 0\\\text{x}+1 &, \text{ if x} > 0\end{cases}\text{at x}=0$
Also sketch the graph of this function.