Trig Integration (1 Viewer)

[Lukybear]

Find the derivative of f(x) = cos x + x sin x. Hence evaluate the definite integral

 

[shaon0]

Find the derivative of f(x) = cos x + x sin x. Hence evaluate the definite integral

f'(x)=-sin(x)+sin(x)+xcos(x)
=xcos(x)
Hence,
S {x=0 to x=pi/2} xcos(x) dx = S {x=0 to x=pi/2} d/dx(cos x + x sin x) dx
= {x=0 to x=pi/2} [(cos x + x sin x)]
= pi/2-1

 

[Lukybear]

How did you integrate the cosx+xsinx?

O crud, nvm

 

[Lukybear]

Also

Derive f(x) = sin2x -2x.cos2x, and us this to find
S{pi/4,0} xsin2x dx

 

[life92]

Derive f(x) = sin2x -2x.cos2x, and us this to find
S{pi/4,0} xsin2x dx

f'(x) = 2cos2x - 2 (cos2x -2xsin2x)
= 4xsin2x

therefore,

S{pi/4,0} xsin2x dx = 1/4 [ sin2x - 2xcos2x ] {pi/4,0}
= 1/4 [ (1 - 0) - (0 - 0)
= 1/4

 

[shaon0]

Also

Derive f(x) = sin2x -2x.cos2x, and us this to find
S{pi/4,0} xsin2x dx

f'(x)=2cos(2x)-2cos2x+4x.sin2x
=4xsin(2x)

S{pi/4,0} xsin2x dx=(1/4)S{pi/4,0}4xsin2x dx
=(1/4) S{pi/4,0} d/dx(sin2x -2x.cos2x) dx
=(1/4) {pi/4,0}sin2x -2x.cos2x
=(1/4)