How do you find the antiderivative of #int 1/(x^2+3x) dx#?
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To find the antiderivative of ( \frac{1}{x^2 + 3x} ) with respect to ( x ), we can use partial fraction decomposition:
[ \frac{1}{x^2 + 3x} = \frac{1}{x(x + 3)} ]
Now, we need to decompose ( \frac{1}{x(x + 3)} ) into partial fractions. We express ( \frac{1}{x(x + 3)} ) as the sum of two fractions with undetermined constants:
[ \frac{1}{x(x + 3)} = \frac{A}{x} + \frac{B}{x + 3} ]
Now, we need to find the values of ( A ) and ( B ). We can do this by finding a common denominator and equating coefficients:
[ 1 = A(x + 3) + Bx ]
Solving for ( A ) and ( B ), we get:
[ A(x + 3) + Bx = Ax + 3A + Bx ] [ 1 = (A + B)x + 3A ]
For this equation to hold true for all ( x ), the coefficients of corresponding powers of ( x ) must be equal. Thus, we have two equations:
- Coefficient of ( x ): ( A + B = 0 )
- Constant term: ( 3A = 1 )
From the second equation, ( A = \frac{1}{3} ). Substituting ( A = \frac{1}{3} ) into the first equation, we find ( B = -\frac{1}{3} ).
Now that we have found ( A ) and ( B ), we can rewrite the original integral as:
[ \int \frac{1}{x^2 + 3x} , dx = \int \left( \frac{1}{3x} - \frac{1}{3(x + 3)} \right) , dx ]
[ = \frac{1}{3} \int \frac{1}{x} , dx - \frac{1}{3} \int \frac{1}{x + 3} , dx ]
Now, integrate each term:
[ = \frac{1}{3} \ln|x| - \frac{1}{3} \ln|x + 3| + C ]
Where ( C ) is the constant of integration.
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When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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