求f(x)=1/(1+sin^x),x∈[0,π]上全体原函数sin^x那个是sin平方x,打错了

Question

求f(x)=1/(1+sin^x),x∈[0,π]上全体原函数
sin^x那个是sin平方x,打错了

Answer 1

f(x)=1/(1+sin^2x)
=1/(3/2-1/2cos2x)
=2/(3-cos2x) (万能公式代换,cos2x=(1-tan^2x)/(1+tan^2x))
=2(1+tan^2x)/(3+3tan^2x-1+tan^2x)
=2sec^2x/(2+4tan^2x)
∫f(x)dx
=∫2sec^2x/(2+4tan^2x)dx
=∫1/(1+2tan^2x)dtanx
=1/2∫1/(1/2+tan^2x)dtanx
=1/2arctan[tanx/(√2/2)]+C