And cos x is the derivative of sin x. Sin x c.
Its done my a method called recognition you merely look at the integral and work backwards saying what function must i differentiate wto get the integral wol ah maths made easy regarding method what tehy tell you.
2sinxcosx integral. 2u du. Using formula for sine of double angle. Integrate 2sin x cos x dxlet u cos x and du sin x dx and pull out the 2 2 integral u du integrate with respect to u 2 u2 2 csimplify u2 creplace u with cos x cos2 x c.
Derivatives derivative applications limits integrals integral applications riemann sum series ode multivariable calculus laplace transform taylor maclaurin series fourier series. The answer above is correct or you can do a u sub. Int2sinxcosx dx int sin2xdx t 2x dt 2dx dx dt 2 1 2int sint dt 1 2 cos t c 1 2 cos 2x c c in rr.
U c. To see these are the same you need another trig identity. You can sign in to vote the answer.
This now becomes integral of 2 sin 2 x cos x. Line equations functions arithmetic comp. 1 2 x integ sin2x dx integ x dx integ sin2x dx xcos2x 4 integ cos2x 4 dx xcos 2x 4 sin2x 8 is the answer.
Which i assume they got from expanding sin2x 2sinxcosx and they then jump to. So xsin2x dx 2. Integral is 2 3 sin 3 x c.
2 sin x cos x dx. U sin x du cos x dx. Now taking integration by parts.
Now take 2sinxcosx sin2x. 2 3 cos x with the limits from above. Line equations functions arithmetic comp.
Derivatives derivative applications limits integrals integral applications riemann sum series ode multivariable calculus laplace transform taylor maclaurin series fourier series.