Question

$x {3} + 2x + 1 \div x {}^{2} + 5x + 6$
In partial fractions ​

Answer 1

Answer:

Find ∫ dx / [(x + 1) (x + 2)]

Answer : The integrand is a proper rational function. Therefore, by using the form of partial fraction from the image above, we have:

1 / [(x + 1) (x + 2)] = A / (x + 1) + B / (x + 2) … (1)

Solving this equation, we get,

A (x + 2) + B (x + 1) = 1

Or, Ax + 2A + Bx + B = 1

x (A + B) + (2A + B) = 1

For LHS to be equal to RHS, we have

A + B = 0 and 2A + B = 1. On solving these two equations, we get

A = 1 and B = – 1.

Therefore, we have

1 / [(x + 1) (x + 2)] = 1 / (x + 1) – 1 / (x + 2)

Hence, ∫ dx / [(x + 1) (x + 2)] = ∫ dx / (x + 1) – ∫ dx / (x + 2)

= log |x + 1| – log |x + 2| + C

Note: Equation (1) is true for all permissible values of x. Some authors use the symbol ‘≡’ to indicate that the statement is an identity and use the symbol ‘=’ to indicate that the statement is an equation, i.e., to indicate that the statement is true only for certain values of x.

Answer 2

Answer:

Refer to the attachment

Hope it helps you

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