Help Solve This Problem Proving That Ok?
the following analysis: X + X ^ 2 - X ^ 3 -1 / (1 + X) ^ 2 when the integral 0 to 1
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tính tích phân sau: X +X^2 - X^3 -1 / (1+X)^2 khi tích phân 0 đến 1
-Anonymous
I = ∫ (x + x² - x³ - 1) / (1 + x)² dx , cận 0 --> 1
Ta có: x + x² - x³ - 1
= -x³ - 2x² - x + 3x² + 6x + 3 - 4x - 4
= -x(x² + 2x + 1) + 3(x² + 2x + 1) - 4(x + 1)
= -x(x + 1)² + 3(x + 1)² - 4(x + 1)
--> (x + x² - x³ - 1) / (1 + x)²
= (-x + 3) - 4/(1 + x)
--> I = ∫ [ (3 - x) - 4/(1 + x) ] dx, cận 0 --> 1
--> I = 3x - x²/2 - 4ln│1 + x│ , cận 0 --> 1
suy ra I = 5/2 - 4ln2
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-Anonymous
Ta có: x + x² - x³ - 1
= -x³ - 2x² - x + 3x² + 6x + 3 - 4x - 4
= -x(x² + 2x + 1) + 3(x² + 2x + 1) - 4(x + 1)
= -x(x + 1)² + 3(x + 1)² - 4(x + 1)
--> (x + x² - x³ - 1) / (1 + x)²
= (-x + 3) - 4/(1 + x)
--> I = ∫ [ (3 - x) - 4/(1 + x) ] dx, cận 0 --> 1
--> I = 3x - x²/2 - 4ln│1 + x│ , cận 0 --> 1
suy ra I = 5/2 - 4ln2
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