∫x^2/(9+4x^2)dx

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∫x^2/(9+4x^2)dx
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∫x^2/(9+4x^2)dx
∫x^2/(9+4x^2)dx

∫x^2/(9+4x^2)dx
∫x^2/(9+4x^2)dx =1/4* ∫4x^2/(9+4x^2)dx
=1/4 * ∫[1-9/(9+4x^2)]dx
=1/4 *∫dx -1/4 * ∫[ 1/(1+(2x/3)^2)]dx
=1/4 *∫dx -1/4 * 3/2* ∫[ 1/(1+(2x/3)^2)]d(2x/3)
=x/4 - 3/8 * arctan(2x/3) + c

求不定积分∫x²/(9+4x²)dx
∫x²/(9+4x²)dx=∫[1/4-9/4(9+4x²)]dx=(1/4)∫dx-(9/4)∫dx/(9+4x²)=x/4-(9/4)∫dx/[9(1+4x²/9)]
=x/4-(1/4)∫dx/[1+(2x/3)²]=x/4-(3/8)∫d(2x/3)/[1+(2x/3)²]=x/4-(3/8)arctan(2x/3)+C

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