Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS5 Marks
Question
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\big(2\log\cos\text{x}-\log\sin2\text{x}\big)\text{dx}$
✓

Answer

We have,
$\text{I}=\int\limits^{\frac{\pi}{2}}_0\big(2\log\cos\text{x}-\log\sin2\text{x}\big)\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_0\big(\log\cos^2\text{x}-\log\sin2\text{x}\big)\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_0\log\frac{\cos^2\text{x}}{\sin\text{x}}\text{ dx}$
$=\int\limits^{\frac{\pi}{2}}_0\log\frac{\cos^2\text{x}}{2\sin\text{x}\cdot\cos\text{x}}\text{ dx}$
$=\int\limits^{\frac{\pi}{2}}_0\log\frac{\cos\text{x}}{2\sin\text{x}}\text{ dx}$
$=\int\limits^{\frac{\pi}{2}}_0\big(\log\cos\text{x}-\log\sin\text{x}-\log2\big)\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_0\log\cos\text{x dx}-\int\limits^{\frac{\pi}{2}}_0\log\sin\text{x dx}-\int\limits^{\frac{\pi}{2}}_0\log2$
We know that
$\int\limits^{\frac{\pi}{2}}_0\log\cos\text{x dx}=\int\limits^{\frac{\pi}{2}}_0\log\sin\text{x dx}\ ...(\text{i})$
Hence from equation (i)
$\text{I}=-\int\limits^{\frac{\pi}{2}}_0\log2=-\frac{\pi}{2}\log2$

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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Verify Rolle's theorem for the following function on the indicated intervals$f(x) = (x^2- 1)(x - 2)$ on $[-1, 2]$
→
Find the equation of the perpendicular drawn from the point P(2, 4, -1) to the line $\frac{\text{x}+5}{1}=\frac{\text{y}+3}{4}=\frac{\text{z}-6}{-9}.$ Also, write down the coordinates of the foot of the perpendicular from P.
→
If the coordinates of the points A, B, C, are $(1, 2, 3), (4, 5, 6), (-4, 3, -6)$ and $(2, 9, 2),$ then find the angle between AB and CD.
→
Differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-\frac{\text{dy}}{\text{dx}}=0,\text{y}(0)=2,\text{y}'(0)=1$Function $\text{y}=\text{e}^\text{x}+1$
→
The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube.
→
$\text{If y = 3} \cos (\log\text{x}) + 4\sin (\log \text{x}), \text{then show that x}^{2} .\frac{\text{d}^{2}{\text{y}}}{\text{dx}^{2}} + \text{y} = 0$
→
Evaluate the following integral:
$\int\frac{1}{\sqrt{(2-\text{x})^2+1}}\text{ dx}$
→
Differentiate the following functions with respect to x:
$\text{x}\sin2\text{x}+5^{\text{x}}+\text{k}^\text{k}+(\tan^2\text{x})^3$
→
Using differentials, find the approximate values of the following:
$\log_\text{e}10.01$ it being given that $\log_{10}=0.4343$
→
Differentiate $(\log\text{x})^\text{x}$ with respect to x.
→