# How do you integrate #(3x+2) / (x^(2)+3x-4)dx# using partial fractions?

or,

To decompose the integrand using Method of Partial Fraction, let,

Therefore,

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To integrate (\frac{{3x + 2}}{{x^2 + 3x - 4}}) using partial fractions, first factor the denominator:

(x^2 + 3x - 4 = (x + 4)(x - 1))

The partial fraction decomposition will have the form:

(\frac{{3x + 2}}{{x^2 + 3x - 4}} = \frac{{A}}{{x + 4}} + \frac{{B}}{{x - 1}})

To find (A) and (B), multiply both sides by the denominator (x^2 + 3x - 4) and simplify:

(3x + 2 = A(x - 1) + B(x + 4))

This gives two equations:

- (3x + 2 = Ax - A + Bx + 4B)
- (3x + 2 = (A + B)x + (4B - A))

By comparing coefficients, we get:

(A + B = 3) (4B - A = 2)

Solve these simultaneous equations to find (A) and (B).

Once you have (A) and (B), integrate each term separately:

(\int \frac{{3x + 2}}{{x^2 + 3x - 4}} dx = \int \frac{{A}}{{x + 4}} dx + \int \frac{{B}}{{x - 1}} dx)

Then integrate each term using the natural logarithm function.

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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.

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