Find integral of (x^3-6x-20)/(x+5) dx
To find the integral of the given function, we can use polynomial long division or synthetic division to divide the numerator polynomial by the denominator polynomial.
Let's perform the division:
x^2 - 5x + 19
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x + 5 | x^3 - 6x - 20
- (x^3 + 5x^2)
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-11x - 20
+ (11x + 55)
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35
The result of the division is a quadratic polynomial, x^2 - 5x + 19, and a remainder of 35.
Therefore, we can rewrite the integral as:
∫ (x^3 - 6x - 20)/(x + 5) dx = ∫ (x^2 - 5x + 19) dx + ∫ (35/(x + 5)) dx
∫ (x^2 - 5x + 19) dx can be easily integrated term by term, resulting in:
= (1/3)x^3 - (5/2)x^2 + 19x + C1, where C1 is the constant of integration.
∫ (35/(x + 5)) dx can be integrated using the natural logarithm:
= 35 ln|x + 5| + C2, where C2 is another constant of integration.
Combining the two integrals, we have:
∫ (x^3 - 6x - 20)/(x + 5) dx = (1/3)x^3 - (5/2)x^2 + 19x + C1 + 35 ln|x + 5| + C2
Hence, the integral of (x^3 - 6x - 20)/(x + 5) dx is (1/3)x^3 - (5/2)x^2 + 19x + 35 ln|x + 5| + C, where C is the constant of integration.