搜搜做题作业网1/((x+1)乘以根号下(x的平方+1))在0到1上的定积分
来源：学生作业帮 编辑：搜搜做题作业网作业帮 分类：数学作业 时间：2024/05/02 03:32:30
1/((x+1)乘以根号下(x的平方+1))在0到1上的定积分
∫(0,1) dx/[(x+1)√(x^2+1)]
令x=tant dx=sec^2tdt
原式=∫(0,π/4) sec^2tdt/[(tant+1)*sect]
=∫(0,π/4) dt/(sint+cost)
=∫(0,π/4) dt/[√2*sin(t+π/4)]
=(1/√2)*∫(0,π/4) csc(t+π/4)*d(t+π/4)
=(1/√2)*ln[csc(t+π/4)-ctg(t+π/4)]|(0,π/4)
=(1/√2)*[ln1-ln(√2-1)]
=(1/√2)*ln(√2+1)
令x=tant dx=sec^2tdt
原式=∫(0,π/4) sec^2tdt/[(tant+1)*sect]
=∫(0,π/4) dt/(sint+cost)
=∫(0,π/4) dt/[√2*sin(t+π/4)]
=(1/√2)*∫(0,π/4) csc(t+π/4)*d(t+π/4)
=(1/√2)*ln[csc(t+π/4)-ctg(t+π/4)]|(0,π/4)
=(1/√2)*[ln1-ln(√2-1)]
=(1/√2)*ln(√2+1)