# How do you integrate #int ( x-5)/(x-2)^2# using partial fractions?

The answer is

By signing up, you agree to our Terms of Service and Privacy Policy

To integrate ( \int \frac{x - 5}{(x - 2)^2} ) using partial fractions, first, express the integrand as the sum of two fractions:

[ \frac{x - 5}{(x - 2)^2} = \frac{A}{x - 2} + \frac{B}{(x - 2)^2} ]

Then, find the values of ( A ) and ( B ) by equating numerators:

[ x - 5 = A(x - 2) + B ]

To solve for ( A ) and ( B ), you can choose suitable values of ( x ) to simplify the equation. For example, choosing ( x = 2 ) eliminates the ( B ) term:

[ 2 - 5 = A(2 - 2) + B ] [ -3 = B ]

Now, for ( A ), choose another suitable value of ( x ), such as ( x = 3 ):

[ 3 - 5 = A(3 - 2) - 3 ] [ -2 = A - 3 ] [ A = 1 ]

Therefore, ( A = 1 ) and ( B = -3 ). Now, rewrite the original integral using the partial fractions:

[ \int \frac{x - 5}{(x - 2)^2} , dx = \int \frac{1}{x - 2} , dx + \int \frac{-3}{(x - 2)^2} , dx ]

These integrals can now be evaluated separately:

[ = \ln|x - 2| - \frac{3}{x - 2} + C ]

Where ( C ) is the constant of integration.

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- How do you integrate #int ln(sint)cost# by integration by parts method?
- How do you find #int (2x)/(x^3-x^2+x-1)dx# using partial fractions?
- How do you integrate #int (3x+4)/(sqrt(3x^2+8x+3))# using substitution?
- How do you integrate #int x arctan x # using integration by parts?
- How do you find the integral of #5/[sqrt(9x^2-16)] dx#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7