考试中急.不定积分Jx方inxdx.J1/((1+x方）x方)dx定积分J上4下01／（2+...考试中急.不定积分Jx方inxdx.J1/((1+x方）x方)dx定积分J上4下01／（2+x的开根）dxJ上3下01／（开根（9-x方））dxJ为那个符

Question

考试中急.不定积分 Jx方inxdx.J1/((1+x方）x方)dx 定积分 J上4下0 1／（2 + ...
考试中急.不定积分
Jx方inxdx.J1/((1+x方）x方)dx
定积分
J上4下0 1／（2 + x的开根）dx
J上3下0 1／（开根（9-x方））dx
J为那个符号,你们懂得.

Answer 1

∫ x²lnx dx
= ∫ lnx d(x³/3)
= (1/3)x³lnx - (1/3)∫ x³*1/x dx
= (1/3)x³lnx - (1/3)*(x³/3) + c
= (1/3)x³lnx - (1/9)x³ + c
= (x³/9)(3lnx - 1) + c
∫ 1/[(1+x²)x²] dx
= ∫ [1/x² - 1/(x²+1)] dx
= -1/x - arctanx + c
∫(0->4) 1/(2+√x) dx
= ∫(0->4) 1/(2+√x) * 2√x / 2√x dx
= 2∫(0->4) √x/(2+√x) d√x
= 2∫(0->4) (2+√x-2)/(2+√x) d√x
= 2∫(0->4) [1 - 2/(2+√x)] d√x
= 2√x|(0->4) - 4ln|2+√x|(4->0)
= 2(2) -4ln(2+2) + 4ln(2)
= 4 - 8ln2 + 4ln2
= 4 - 4ln2
∫(0->3) 1/√(9-x²) dx
令x = 3sinθ,dx = 3cosθ dθ
当x = 0,θ = 0；当x = 3,θ = π/2
= ∫(0->π/2) (3cosθ)/(3cosθ) dθ
= θ(0->π/2)
= π/2
希望满意!