Question

求解3道不定积分的问题 谢谢了，最好有过程

sin3x*sin5xdx
x^5/(1+x^3)
1-x^7/x(1+x^7)

Answer 1

sin3x sin5x = 1/2 [cos(-2x) - cos(8x)] = 1/2 cos2x - 1/2 cos8x
∫ sin3x sin5x dx = -1/2 ∫ cos2x dx - 1/2 ∫ cos8x dx
= (1/4)sin2x - (1/16)sin8x + C
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∫ x^5/(1 + x³) dx
= ∫ x²(x³ + 1 - 1)/(1 + x³) dx
= ∫ x² dx - ∫ x²/(1 + x³) dx
= x³/3 - ∫ 1/(1 + x³) d(x³/3)
= x³/3 - (1/3)ln|1 + x³| + C
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∫ (1 - x^7)/[x(1 + x^7)] dx
= ∫ [(1 + x^7) - 2x^7]/[x(1 + x^7)] dx
= ∫ 1/x dx - 2∫ x^6/(1 + x^7) dx
= ∫ 1/x dx - (2/7)∫ d(x^7)/(1 + x^7)
= ln|x| - (2/7)ln|1 + x^7| + C

Answer 2

∫ sin3x*sin5x dx = ∫ -½ (cos8x-cos2x) dx = ¼ sin(2x) - 1/16 sin(8x) + C

∫ x^5/(1+x³) dx
=∫ (x^5 + x² - x²)/(1+x³) dx
=∫ [x² - x²/(1+x³)]dx
=∫ x²dx - 1/3 ∫ 1 /(1+x³) d(1+x³)
= 1/3 x³ - 1/3 ln|1+x³| + C

∫ (1-x^7) / [x(1+x^7)] dx
=∫ (1+x^7-2 x^7) / [x(1+x^7)] dx
=∫ [1/x - 2 x^6 / (1+x^7)] dx
=ln|x| - 2/7 ln|1+x^7| + C