Integrate this

Calculus Level 3

Let [a] denote the greatest integer which is less than or equal to a. Then the value of the given integration is ∫ − π / 2 π / 2 [ S i n x C o s x ] d x \int _{ -\pi /2 }^{ \pi /2 }{ [Sinx Cos x } ]dx

π / 2 \pi/2 π \pi − π / 2 -\pi/2 − π -\pi

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Suyash Mittal by Suyash Mittal , 24, India

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1 solution

Suyash Mittal
Apr 24, 2014

I = ∫ − π / 2 π / 2 [ 1 2 S i n 2 x ] d x I\quad =\quad \int _{ -\pi /2 }^{ \pi /2 }{ [\frac { 1 }{ 2 } } Sin2x]dx Put 2 x = θ 2x = \theta or, 2 d x = d θ 2dx = d\theta I = 1 2 ∫ − π π [ 1 2 S i n θ ] d θ = 1 2 [ ∫ − π 0 ( − 1 ) d x + ∫ 0 π 0 d x ] I\quad =\quad \frac { 1 }{ 2 } \int _{ -\pi }^{ \pi }{ [\frac { 1 }{ 2 } } Sin\theta ]d\theta \quad =\quad \frac { 1 }{ 2 } [\int _{ -\pi }^{ 0 }{ (-1)dx\quad +\quad \int _{ 0 }^{ \pi }{ 0\quad dx] } } = 1 2 ( − ) ( x ) − π 0 + 0 = − 1 2 ( 0 + π ) = − π 2 =\frac { 1 }{ 2 } (-){ (x) }_{ -\pi }^{ 0 }\quad +\quad 0\quad =\quad -\frac { 1 }{ 2 } (0+\pi )\quad =\quad -\frac { \pi }{ 2 }

4 Helpful 0 Interesting 1 Brilliant 0 Confused

Even though I clicked − π 2 -\frac{\pi}{2} , it displayed that I clicked π 2 \frac{\pi}{2} .

Anish Puthuraya - 7 years, 1 month ago

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Might be some kind of bug

Suyash Mittal - 7 years, 1 month ago

how could you change the integrand in the second last step

Rajdeep Dhingra - 6 years, 4 months ago

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