Advertisements
Advertisements
Question
Options
π
π/2
0
2π
Advertisements
Solution
0
\[I = \int_0^\frac{\pi}{2} \sin2x \log \tan x\ d x . . . . . \left( 1 \right)\]
\[I = \int_0^\frac{\pi}{2} \sin\left( \pi - 2x \right) \log \tan\left( \frac{\pi}{2} - x \right) d x\]
\[I = \int_0^\frac{\pi}{2} \sin2x \log \cot x\ d x . . . . . \left( 2 \right)\]
\[\text{Adding} \left( 1 \right) and \left( 2 \right), \text{we get}, \]
\[2I = \int_0^\frac{\pi}{2} \sin2x\left( \log \tan x + \log \cot x \right) d x\]
\[2I = \int_0^\frac{\pi}{2} \sin2x\left( \log \tan x \cot x \right) d x\]
\[2I = \int_0^\frac{\pi}{2} \sin2x\left( \log1 \right) d x\]
\[I = 0\]
APPEARS IN
RELATED QUESTIONS
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that \[\int_a^b xf\left( x \right)dx = \frac{a + b}{2} \int_a^b f\left( x \right)dx\]
The value of \[\int\limits_0^{\pi/2} \cos x\ e^{\sin x}\ dx\] is
The derivative of \[f\left( x \right) = \int\limits_{x^2}^{x^3} \frac{1}{\log_e t} dt, \left( x > 0 \right),\] is
If f (a + b − x) = f (x), then \[\int\limits_a^b\] x f (x) dx is equal to
\[\int\limits_0^1 \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) dx\]
\[\int\limits_{\pi/3}^{\pi/2} \frac{\sqrt{1 + \cos x}}{\left( 1 - \cos x \right)^{5/2}} dx\]
\[\int\limits_0^{\pi/4} \tan^4 x dx\]
\[\int\limits_0^\pi \frac{x}{a^2 \cos^2 x + b^2 \sin^2 x} dx\]
\[\int\limits_{- \pi/4}^{\pi/4} \left| \tan x \right| dx\]
\[\int\limits_0^{15} \left[ x^2 \right] dx\]
\[\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} dx\]
Prove that `int_a^b ƒ ("x") d"x" = int_a^bƒ(a + b - "x") d"x" and "hence evaluate" int_(π/6)^(π/3) (d"x")/(1+sqrt(tan "x")`
Evaluate the following using properties of definite integral:
`int_0^(i/2) (sin^7x)/(sin^7x + cos^7x) "d"x`
Choose the correct alternative:
Γ(n) is
Find `int x^2/(x^4 + 3x^2 + 2) "d"x`
Find `int sqrt(10 - 4x + 4x^2) "d"x`
If x = `int_0^y "dt"/sqrt(1 + 9"t"^2)` and `("d"^2y)/("d"x^2)` = ay, then a equal to ______.
`int (x + 3)/(x + 4)^2 "e"^x "d"x` = ______.
Evaluate \[\int_{0}^{1}x\,dx\].
What is an antiderivative of \[2x+1\]?
