Use substitution to evaluate the indefinite integral:
The integral of [ (sq. root (1 + ln x)) ((ln x)/x) dx]
Im confused on what i should substitute u and du for.
Thank you so much!!
Let u = 1+ln x
so,
du = 1/x dx
ln x = u-1
So, we have
u^1/2 * (u-1) du
= u^3/2 - u^1/2 du
Integrate that to get
2/5 u^5/2 - 2/3 u^3/2
=
2/5 (1 + ln x)^5/2 - 2/3 (1+ln x)^3/2
Check your work by taking the derivative. Trust me -- it comes out right.
To evaluate the given integral using substitution, we need to choose an appropriate substitution. Let's consider the expression inside the integral:
√(1 + ln(x)) * (ln(x)/x) dx
Here, we can let u = 1 + ln(x). So, our substitution will be:
u = 1 + ln(x)
To find du, we differentiate both sides of the equation with respect to x:
du/dx = d(1 + ln(x))/dx
= 0 + (1/x)
= 1/x
To find dx, we can solve for it in terms of du:
du = (1/x) dx
Now, we have expressed dx in terms of du, so we can rewrite the integral using our substitution:
∫ [√u * (ln(x)/x)] dx
= ∫ √u * (ln(x)/x) d(x)
= ∫ √u * (ln(x)/x) (1/x) du
= ∫ √u * ln(x) * (1/x^2) du
Now, we can simplify this expression and integrate with respect to u.