integration of
[tex] \\ 1 \\ 6 + 5x - 6x {}^{2} [/tex]
please use partial function
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integration of
[tex] \\ 1 \\ 6 + 5x - 6x {}^{2} [/tex]
please use partial function
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Answer:
The integral of the given expression can be computed using partial fraction decomposition. Here's how you can do it:
∫(1/6 + 5x - 6x^2) dx
First, factor the quadratic term:
∫(-6x^2 + 5x + 1/6) dx
Now, we need to express the integrand as a sum of partial fractions. The denominator has three distinct linear factors, so we can write:
-6x^2 + 5x + 1/6 = A/x + B/(2x - 1) + C/(3x + 2)
Next, find the values of A, B, and C. You can do this by finding a common denominator and matching coefficients:
-6x^2 + 5x + 1/6 = (A(2x - 1)(3x + 2) + B(x)(3x + 2) + C(x)(2x - 1))/(x(2x - 1)(3x + 2))
Now, equate the numerators:
-6x^2 + 5x + 1/6 = (A(2x - 1)(3x + 2) + B(x)(3x + 2) + C(x)(2x - 1))
You can solve for A, B, and C by comparing coefficients. Once you find those values, you can integrate each term separately. The integral of the sum of the partial fractions will give you the integral of the original expression.
Please note that finding the values of A, B, and C involves some algebraic manipulations. After you find these values, you can proceed to compute the integrals of the individual terms.