设G（x）=定积分（上下线分别为x^2,1)cost*dt/t,求定积分（上下线分别为1,0)xG（x）dx

Question

设 G（x）=定积分（上下线分别为x^2,1)cos t * dt/t,求 定积分（上下线分别为1,0)xG（x）dx

Answer 1

G(x) = ∫[1→x²] cost/t dt
G'(x) = dx²/dx * cos(x²)/x²
= 2x * cos(x²)/x²
= 2cos(x²)/x
注意G(1) = ∫[1→1] cost/t dt = 0
∴
∫[0→1] xG(x) dx
= ∫[0→1] G(x) d(x²/2)
= (1/2)x²G(x) |[0→1] - (1/2)∫[0→1] x² d[G(x)]、这里运用了分部积分法
= (1/2)[(1)²G(1) - (0)²G(0)] - (1/2)∫[0→1] x²G'(x) dx、将上面G'(x)的结果代入这里
= (1/2)[(1)²(0) - (0)²G(0)] - (1/2)∫[0→1] x² * 2cos(x²)/x dx
= - ∫[0→1] xcos(x²) dx
= - ∫[0→1] cos(x²) d(x²/2)、凑微分
= (- 1/2)∫[0→1] cos(x²) d(x²)
= (- 1/2)[sin(x²)] |[0→1]
= (- 1/2)[sin(1²) - sin(0²)]
= (- 1/2)sin(1)