∫1/(1+e^x)dx
=-∫e^x/(1+e^x)d[e^(-x)]
=-∫1/[1+e^(-x)]d[e^(-x)]
=-∫1/[1+e^(-x)]d[1+e^(-x)]
=-ln[1+e^(-x)]+C
∫1/(4-x²)dx
=∫1/[(2+x)(2-x)]dx
=(1/4)∫1/(2-x)+1/(2+x)]dx
=(1/4)[ln(2+x)-ln(2-x)]+C
=(1/4)ln[(2+x)/(2-x)]+C
∫sin³xdx
=-∫(1-cos²x)dcosx
=-cosx+cos³x/3+C
∫(cscx)^4dx
=∫csc²xdctanx
=∫(1+ctan²x)dctanx
=ctanx+ctan³x/3+C
∫1/[x+x^(2/3)]dx
=∫1/{x^(2/3)[1+x^(1/3)]}dx
=3∫1/[1+x^(1/3)]dx^(1/3)
=2ln[1+x^(1/3)]+C
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