用第一类换元积分法求下列不定积分⑴∫cos4xdx;⑵∫e^-2xdx;⑶∫(3x－1)^4dx;⑷∫10^3xdx

Question

用第一类换元积分法求下列不定积分
⑴∫cos4x dx; ⑵∫e^-2x dx; ⑶∫ (3x－1)^4 dx; ⑷∫10^3x dx

Answer 1

(1)令u = 4x,du = 4dx
∫ cos4x dx
= (1/4)∫ cosu du
= (1/4)sinu + C
= (1/4)sin(4x) + C
————————————————————————————
(2)令u = - 2x,du = - 2 dx
∫ e^(- 2x) dx
= (- 1/2)∫ e^u du
= (- 1/2)e^u + C
= (- 1/2)e^(- 2x) + C
= - 1/[2e^(2x)] + C
————————————————————————————
(3)令u = 3x - 1,du = 3 dx
∫ (3x - 1)^4 dx
= (1/3)∫ u^4 du
= (1/3)(1/5)u^5 + C
= (1/15)(3x - 1)^5 + C
————————————————————————————
(4)令u = 3x,du = 3 dx
∫ 10^(3x) dx
= (1/3)∫ 10^u du
= (1/3) * (10^u)/ln(10) + C
= (10^u)/[3ln(10)] + C
= [10^(3x)]/[3ln(10)] + C