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We need to find the definite integral of y = (1/x)*(7+ln x)^2.
Let u = 7 + ln x
du/dx = 1/x
=> dx / x = du
Int [ y dx] = Int [ (1/x)*(7+ln x)^2 dx ]
=> Int [ u^2 du]
=> u^3 / 3
replace u with 7 + ln x
=> (7 + ln x)^3 / 3
Therefore the integral of y = (1/x)*(7 + ln x)^2 is
(7 + ln x)^3 / 3 + C
To find the indefinite integral of y=(1/x)*(7+lnx)^2
We put (7+ln x) = t, then by differentiating (y+lnx) we get: (1/x) dx = dt
So thre goven integral is rewritten as:
Int y dx = Int (1/x)*(1+lnx)^2 dx = Int (7+lnx)*(1/x) dx = Int t^2 dt = (t^(2+1))/(2+1) + constant.
=> Int (1/x)*(1+lnx)^2 dx = (t^3)/3 +C
Now we replace t = (7+lnx).
So Integral (1/x)*(1+lnx)^2 dx = (1/3)(7+lnx)^3 + C.
We'll apply the substitution technique to evaluate the indefinite integral of the given function.
Int f(x)dx = Int (7+ln x)^2dx/x
We'll substitute 7 + ln x = t.
We'll differentiate both sides:
dx/x = dt
We'll re-write the integral, having t as variable:
Int t^2 dt = t^3/3 + C
But t = 7 + ln x
The answer is: Int (7+ln x)^2dx/x = (7 + ln x)^3/3 + C
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