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∫[0,ln5) e^x√(e^x-1)dx/(e^x+3)
=∫[0,ln5)√(e^x-1)de^x /(e^x+3)
e^x=u x=0,u=1 x=ln5 u=5
=∫[1,5]√(u-1)du/(u+3)
=∫[1,5]2(u-1)d√(u-1)/(u+3)
=∫[1,5]2d√(u-1) +∫[1,5] (-4d√(u-1)/(u+3))
=2*2+∫[1,5](-4) d√(u-1)/[√(u-1)^2+4]
=4 -∫[1,5]d√(u-1)/[√(u-1)^2/4+1]
=4-2arctan(√(u-1)/2)|[1,5]
=4-2*(π/4)
=4-π/2
=∫[0,ln5)√(e^x-1)de^x /(e^x+3)
e^x=u x=0,u=1 x=ln5 u=5
=∫[1,5]√(u-1)du/(u+3)
=∫[1,5]2(u-1)d√(u-1)/(u+3)
=∫[1,5]2d√(u-1) +∫[1,5] (-4d√(u-1)/(u+3))
=2*2+∫[1,5](-4) d√(u-1)/[√(u-1)^2+4]
=4 -∫[1,5]d√(u-1)/[√(u-1)^2/4+1]
=4-2arctan(√(u-1)/2)|[1,5]
=4-2*(π/4)
=4-π/2
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