∫4 [( ( ln(z) )^ 2)/z] dz

∫[( 7 e^(7 √x))/ √x] dx

∫[(2 x + 5)/( x^2 + 5 x + 2)] dx

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Let x^2 +5x +2 = u

That makes du = (2x +5) dx
So your integral becomes du/u = ln u = ln (x^2 + 5x + 2)

Try a similar substitution technique with the others.

To find the solution to each of these integrals, we can use different integration techniques.

1. ∫4 [( ln(z) )^2/z] dz:
We can begin by substituting a new variable to simplify the integral. Let u = ln(z). Then, du/dz = 1/z, and dz = z du.
By substituting these values, the integral becomes: ∫4 [u^2] z du.
Next, we can simplify the expression further by pulling the constant factor out of the integral: 4 ∫u^2 z du.
Now, we can integrate u^2 with respect to u: 4 ∫u^2 du = 4 (u^3/3) + C.
Finally, substitute the original variable back in: (4/3) ln(z)^3 + C, where C is the constant of integration.

2. ∫[7 e^(7 √x)/√x] dx:
This integral involves a combination of exponential and square root functions. We can use the technique of substitution to simplify it.
Let u = √x, then du/dx = 1/(2√x), and dx = 2√x du.
By substituting these values, the integral becomes: ∫[7 e^(7u) / u] (2√x du).
Simplifying further: 14 ∫e^(7u) du.
Now, we can integrate e^(7u) with respect to u: (14/7) e^(7u) + C.
Finally, substitute the original variable back in: (14/7) e^(7√x) + C, where C is the constant of integration.

3. ∫[(2x + 5) / (x^2 + 5x + 2)] dx:
This integral involves a rational function. We can use the method of partial fractions to solve it.
First, factor the denominator of the rational function: x^2 + 5x + 2 = (x+1)(x+2).
Now, we can express the original rational function as the sum of two fractions, with undetermined coefficients A and B:
(2x + 5) / (x^2 + 5x + 2) = A / (x+1) + B / (x+2).
To find the values of A and B, we can multiply both sides by the denominator: (2x + 5) = A(x+2) + B(x+1).
Expanding the equation: 2x + 5 = (A + B) x + (2A + B).
Equating the coefficients of matching powers of x, we get:
2 = A + B --> Equation 1
5 = 2A + B --> Equation 2
Solving these equations simultaneously will help us find the values of A and B.
By solving Equation 1, we find that A = 3.
Substituting the value of A in Equation 2, we get: 5 = 6 + B. Therefore, B = -1.
So, we can rewrite the integral as: ∫[3 / (x+1) - 1 / (x+2)] dx.
And now, we can integrate each term separately:
∫[3 / (x+1)] dx = 3 ln|x+1| + C1, where C1 is the constant of integration.
∫[-1 / (x+2)] dx = -ln|x+2| + C2, where C2 is the constant of integration.
Combining the results, the final solution is: 3 ln|x+1| - ln|x+2| + C, where C = C1 - C2 is the constant of integration.