Question

Wyznacz całkę nieoznaczoną funkcji

Answer 1

Sinx=[2sinx/2*cosx/2]/(sin²x/2+cos²x/2)=2t(1+t²)
cosx=[cos²x/2-sin²x/2]/(sin²x/2+cos²x/2)=(1-t²)/(1+t²)
dgdzie t=tgx/2⇒x/2=arctg(t)
dx=2/(1+t²)·dt
sinx-cosx+√2=(2t+t²-1+√2+t²√2)/(1+t²)
dx/(sinx-cosx+√2)=2dt/[2t+t²(1+√2)-1]
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2-gi sposob
sinx-cosx=sinx-sin(π/2-x)=2sin(x-π/4)·cos(π/4)=√2·sin(x-π/4)
u=x-π/4