Question

高数（分部积分法）问题= =1 ∫e^x乘以(cosx)^2 dx2 ∫e^arcsinx dx3 ∫(sinx)^6

Answer 1

1.cos²x = 1/2 + cos(2x) /2

∫ e^x [1/2 + cos(2x) /2] dx = (1/2) e^x + (1/2) ∫ e^x cos(2x) dx

I = ∫ e^x cos(2x) dx = ∫ cos(2x) d（e^x ) = e^x cos(2x) - ∫ e^x [-2 sin(2x)] dx

= e^x cos(2x) + ∫ 2 sin(2x) d(e^x)

= e^x cos(2x) + 2e^x sin(2x) - 4 ∫ e^x cos(2x) dx

=> I = (1/5) [e^x cos(2x) + 2e^x sin(2x) ] + C

原式 = (1/2) e^x + (1/10) [e^x cos(2x) + 2e^x sin(2x) ] + C

2.I = x * e^arcsinx - ∫ e^arcsinx * x/√(1-x²) dx = x * e^arcsinx + ∫ e^arcsinx d√(1-x²)

= x * e^arcsinx + √(1-x²) e^arcsinx - I

=> I = (1/2) e^arcsinx [ x + √(1-x²) ] + C

3.∫ [-π/2,π/2] (sinx)^6 dx = 2 ∫ [0,π/2] (sinx)^6 dx 有递推公式

= 2 * [ 5*3*1 /(6*4*2)] * π/2

= π/16