How do you integrate #1 / [ (12x)(1x) ]# using partial fractions?
The answer is
Now let's begin breaking down into partial fractions.
We compare the numerators since the denominators are the same.
Consequently,
so,
By signing up, you agree to our Terms of Service and Privacy Policy
To integrate ( \frac{1}{(12x)(1x)} ) using partial fractions, follow these steps:
 Decompose the fraction into partial fractions.
 Find the constants for each partial fraction.
 Integrate each partial fraction separately.
The steps:

Decompose ( \frac{1}{(12x)(1x)} ) into partial fractions. [ \frac{1}{(12x)(1x)} = \frac{A}{12x} + \frac{B}{1x} ]

Find the constants ( A ) and ( B ). [ 1 = A(1x) + B(12x) ]

Solve for ( A ) and ( B ) by equating coefficients of like terms. [ 1 = (A + B)  (A + 2B)x ]

Equate coefficients:
 Constant term: ( A + B = 1 )
 Coefficient of ( x ): ( A  2B = 0 )

Solve the system of equations for ( A ) and ( B ). [ A = \frac{2}{3}, \quad B = \frac{1}{3} ]

Rewrite the original integral using the partial fractions: [ \int \frac{1}{(12x)(1x)} dx = \int \left( \frac{2}{3(12x)} + \frac{1}{3(1x)} \right) dx ]

Integrate each partial fraction separately: [ = \frac{2}{3} \int \frac{1}{12x} dx + \frac{1}{3} \int \frac{1}{1x} dx ]

Perform the integrals: [ = \frac{1}{3} \ln12x  \frac{1}{3} \ln1x + C ] [ = \frac{1}{3} \ln12x(1x) + C ]
So, the integral of ( \frac{1}{(12x)(1x)} ) using partial fractions is ( \frac{1}{3} \ln12x1x + C ), where ( C ) is the constant of integration.
By signing up, you agree to our Terms of Service and Privacy Policy
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.
 How do you integrate #int (4x)/sqrt(x^214x+53)dx# using trigonometric substitution?
 How do you integrate by substitution #int u^2sqrt(u^3+2)du#?
 How do you integrate #(4x^3+2x^2+1)/(4x^3x)# using partial fractions?
 How do you express as a partial fraction #1/(s^2 +1)^2#?
 How do you integrate #int sqrt(1x^2)# by trigonometric substitution?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7