S(x²(1-x²)¹/²)dx(adsbygoogle = window.adsbygoogle || []).push({});

without use x=sen(T)?

Sx^2(1-x^2)^1/2dx

x=sen(t)

dx=cos(t)dt (-pi/2<=t<=pi/2)

S(x^2(1-x^2)^1/2)dx=S(sen^2(t)cos^2(t))dt

S(sen^2(t)cos^2(t)^)dt=S((1-cos^2(2t))/4)dt

=S1/4dt - S((co^2(2t))/4)dt= t + c1 - S((1-cos(4t))/8)dt

=t/4 + c1 - S1/8dt + S(cos(4t)/8)dt= t/4 + c1 - t/8 + sen(4t)/32 + c2

=t/8 + sen(4t)/32 + C

but:

sen(4t)=2sen(2t)cos(2t)=2(2sentcost)(1-2sen^2(t))

=4sentcost-8sen^3(t)cost

and t=arcsen(x)

logo

Sx^2(1-x^2)^1/2dx= 1/8arcsen(x)+ 1/8x(1-x^2)^1/2 - 1/4x^3(1-x^2)^1/2 + C

but show that D(1/8arcsen(x)+ 1/8x(1-x^2)^1/2 - 1/4x^3(1-x^2)^1/2 + C)=x^2(1-x^2)^1/2 is another history!is a very hard work? someone agree with me?

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# How to integrate without trigonometric substitution

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