Integrate:(K+1/k^2+4k+8)dx
To integrate the given expression, (K+1)/(k^2+4k+8), with respect to x, we can first simplify it by factoring the denominator:
k^2 + 4k + 8 = (k + 2)^2 + 4
Now, we can rewrite the expression as:
(K + 1) / ((k + 2)^2 + 4)
To integrate this expression, we can use a substitution. Let's make the substitution:
u = k + 2
Now, we need to find the derivative of u to find the value of dx:
du/dk = 1
dx = du
Substituting the value of u and dx in the integral, we get:
∫(K + 1) / ((k + 2)^2 + 4) dx = ∫(K + 1) / (u^2 + 4) du
This integral can be solved using partial fraction decomposition. The denominator in the form (u^2 + a^2) can be written as:
(u^2 + a^2) = (u + ai)(u - ai)
For our expression, a = 2, so we can express the denominator as:
(u^2 + 4) = (u + 2i)(u - 2i)
Now, we need to express the numerator as a sum of fractions with the factors (u + 2i) and (u - 2i) in the denominators:
(K + 1) = A(u + 2i) + B(u - 2i)
Multiplying through by (u + 2i)(u - 2i), we get:
(K + 1) = A(u - 2i)(u + 2i) + B(u + 2i)(u - 2i)
(K + 1) = A(u^2 - (2i)^2) + B(u^2 - (2i)^2)
(K + 1) = (A + B)u^2 - 4i^2(A + B)
(K + 1) = (A + B)u^2 + 4(A + B)
Comparing the coefficient of u^2 and the constant term on both sides, we get:
A + B = 0 (coefficient of u^2)
4(A + B) = K + 1 (constant term)
Solving these equations, we find:
A = B = 1/4
Now, we can rewrite the integral as:
∫(K + 1) / ((k + 2)^2 + 4) dx = ∫(1/4) * [(u + 2i)/(u^2 + 4)] + (1/4) * [(u - 2i)/(u^2 + 4)] du
The integral of (u + 2i)/(u^2 + 4) can be evaluated as:
(1/4) * ln(u^2 + 4)
And the integral of (u - 2i)/(u^2 + 4) can be evaluated as:
(1/4) * 2i * arctan(u/2)
Substituting the values of u back in terms of k, we finally have the result:
∫(K + 1) / ((k + 2)^2 + 4) dx = (1/4) * ln((k + 2)^2 + 4) + (1/4) * 2i * arctan((k + 2)/2) + C
where C is the constant of integration.