∫1/x(x^6 1)dx
=∫(1/[1/t(1/t^6 1)]d(1/t)
=∫(t^7*(-1/t^2)/(1 t^6)dt
=-∫t^5/(1 t^6)dt
=-1/6∫1/(1 t^6)d(1 t^6)
=-1/6ln|1 t^6| >
Al poner T=1/x, la fórmula original =-1/6ln | 1 1/x 6 | p>
2.
sin(A)cos(B)=(1/2)[sin(A B) sin(A-B)]
∴sin(2x)cos(3x)=(1/2)[ sin(2x 3x) sin(2x-3x)]
= (1/2)[sin5x sin(-x)]
= (1/2)(sin5x - sinx)
∴∫ sin(2x)cos(3x) dx
=(1/2)∫sin(5x)dx-(1/2)∫sinx dx
=(1/2)(1/5)∫sin(5x)d(5x)-(1/2)∫sinx dx
=(1/10)(- cos(5x)] (1/2)cosx C
=(1/10)[5 cosx-cos(5x)] C
4.=∫(1/x ^4-1/ x^6)dx=-1/3*x^(-3) 1/5x^(-5) c
5.∫(lnx)^3/x^2dx
=-∫(lnx)^3(1/x)'dx
=-(lnx)^3(1/x) 3∫(lnx)^2(1 /x)^2dx
=-(lnx)^3(1/x)-3∫(lnx)^2(1/x)'dx
=-(lnx )^3/x -3(lnx)^2/x 6∫lnx(1/x)^2dx
=-(lnx)^3/x-3(lnx)^2/x- 6lnx/x ∫1 /x^2dx
=-[(lnx)^3 3(lnx)^2 6lnx 1](1/x) c.