How do you find the indefinite integral of #int (x^23x^2+5)/(x3)#?
Start by simplifying the numerator.
#=>int(5  2x^2)/(x  3)# We now divide
#5  2x^2# by#x 3# .
Hence, our integral becomes
#int(2x  6  13/(x  3))# , which can be integrated using the rule#int(x^n)dx = x^(n + 1)/(n + 1) + C# and the rule#int(1/u)du = lnu + C# .
#=>x^2  6x  13lnx  3 + C# Hopefully this helps!
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To find the indefinite integral of (\int \frac{x^2  3x^2 + 5}{x  3}), you can use polynomial long division or the method of partial fractions.

Polynomial Long Division: [ \begin{array}{rll} & x & 3 \ \hline x^2  3x^2 + 5 & x^2  3x^2 & + 5 \ & x^2 & 3x \ \hline & & 2x & + 5 \ \end{array} ] So, (\frac{x^2  3x^2 + 5}{x  3} = x  2) with a remainder of (5). Therefore, the integral becomes: [ \int \frac{x^2  3x^2 + 5}{x  3} , dx = \int (x  2) , dx + \int \frac{5}{x  3} , dx ] [ = \frac{x^2}{2}  2x + 5 \lnx  3 + C ]

Method of Partial Fractions: Factorizing the denominator (x  3), we get: [ \frac{x^2  3x^2 + 5}{x  3} = \frac{x^2  3x^2 + 5}{(x  3)} = \frac{(x^2  3x^2 + 5)}{(x  3)} = \frac{(x^2  3x^2 + 5)}{(x  3)} = \frac{x^2  3x^2 + 5}{x  3} ] [ = \frac{(x^2 + 5)}{(x  3)}  \frac{3x^2}{(x  3)} ] [ = (x + 2)  3\frac{x^2}{(x  3)} ] Integrating term by term, we get: [ \int \frac{x^2  3x^2 + 5}{x  3} , dx = \int (x + 2) , dx  3\int \frac{x^2}{x  3} , dx ] [ = \frac{x^2}{2} + 2x  3\left(\frac{x^2}{2}  3x + 9 \lnx  3\right) + C ] [ = \frac{x^2}{2} + 2x  \frac{3}{2}x^2 + 9x  27\lnx  3 + C ] [ = \frac{1}{2}x^2 + 11x  27\lnx  3 + C ]
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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