Question

计算积分😢

Answer 1

令x=tant，dx=sec^2tdt
原式=∫(0,π/4) ln(1+tant)/sec^2t*sec^2tdt
=∫(0,π/4) ln(1+tant)dt
=∫(0,π/4) ln[(cost+sint)/cost]dt
=∫(0,π/4) ln(cost+sint)dt-∫郑拆(0,π/4) ln(cost)dt
=∫(0,π/4) ln[√2*cos(t-π/4)]dt-∫(0,π/4) ln(cost)dt
=∫(0,π/4) ln(√2)dt+∫(0,π/4) ln[cos(π/4-t)]dt-∫(0,π/4) ln(cost)dt
针对中衫肆间那个定或丛轿积分，令u=π/4-t，dt=-du
原式=ln(√2)t|(0,π/4)-∫(π/4,0) ln(cosu)du-∫(0,π/4) ln(cost)dt
=(ln2)π/8

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谢谢！😄

Answer 2

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谢谢！前面有个学霸先回答了，一个小题就这么难啊，谢谢你的帮助！