计算这个积分
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令x=tant，则dx=sec^2tdt
原式=∫(0,π/2) tsec^2t/sec^5tdt
=∫(0,π/2) tcos^3tdt
=(1/4)*∫(0,π/2) t(cos3t+3cost)dt
=(1/4)*∫(0,π/2) tcos3tdt+(3/4)*∫(0,π/2) tcostdt
=(1/12)*∫(0,π/2) td(sin3t)+(3/4)*∫(0,π/2) td(sint)
=(1/12)*tsin3t|(0,π/2)-(1/12)*∫(0,π/2) sin3tdt+(3/4)*tsint|(0,π/2)-(3/4)*∫(0,π/2) sintdt
=-π/24+(1/36)*cos3t|(0,π/2)+3π/8+(3/4)*cost|(0,π/2)
=π/3-1/36-3/4
=π/3-7/9
原式=∫(0,π/2) tsec^2t/sec^5tdt
=∫(0,π/2) tcos^3tdt
=(1/4)*∫(0,π/2) t(cos3t+3cost)dt
=(1/4)*∫(0,π/2) tcos3tdt+(3/4)*∫(0,π/2) tcostdt
=(1/12)*∫(0,π/2) td(sin3t)+(3/4)*∫(0,π/2) td(sint)
=(1/12)*tsin3t|(0,π/2)-(1/12)*∫(0,π/2) sin3tdt+(3/4)*tsint|(0,π/2)-(3/4)*∫(0,π/2) sintdt
=-π/24+(1/36)*cos3t|(0,π/2)+3π/8+(3/4)*cost|(0,π/2)
=π/3-1/36-3/4
=π/3-7/9
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