求下列不定积分 ∫根号X（X-3）dx ∫(x^2+根号下x^3+3)/根号X dx ∫(e^2t)-1/(e^t)-1 dt

Question

要过程。

Answer 1

∫ √x(x - 3) dx
= ∫ [x^(3/2) - 3√x] dx
= (2/5)x^(5/2) - 2x^(3/2) + C

∫ (x² + √x³ + 3)/√x dx
= ∫ x^(2 - 1/2) dx + ∫ x^(3/2 - 1/2) dx + 3∫ x^(- 1/2) dx
= ∫ x^(3/2) dx + ∫ x dx + 3∫ x^(- 1/2) dx
= (2/5)x^(3/2) + x^2/2 + 6√x + C

∫ [e^(2t) - 1]/(e^t - 1) dt
= ∫ [(e^t + 1)(e^t - 1)]/(e^t - 1) dt
= ∫ (e^t + 1) dt
= e^t + t + C