Evaluate the indefinite integral.
(5dx)/(xln(2x))
First make the substitution:
t=ln(2x)=ln2+lnx
dt=d(ln2)/dx + dx/x = dx/x
I = ∫5dx/(xln(2x))
=5∫(dx/x)/ln(2x)
=5∫(dt/t)
=5ln(t) + C
=5ln(ln(2x)) + C
Check by differentiation of I.
That's what I got too, but it says that it's wrong.
I have differentiated the result and got back the integrand.
So possible problems could be:
1. Recheck the expression of the integrand, which contains sufficient parentheses to render it inambiguous. But do check if there are incorrect parentheses or powers.
2. Since you know the result is "wrong" without knowing the actual answer, I assume you are dealing with a software, which is notorious for rejecting correct answers presented in a different format.
Read through the directives carefully, sometimes they specify the name of the integration constant to be other than "C", or log in place of ln, etc.
To evaluate the indefinite integral ∫(5dx)/(xln(2x)) , we can use a technique called substitution. The goal of substitution is to simplify the integrand by substituting a new variable in place of the original one.
Let's proceed step-by-step:
Step 1: Choose a substitution.
In this case, we can let u = ln(2x). This choice will simplify the integrand since we have 1/(xln(2x)) in our original integral.
Step 2: Calculate du/dx.
To find du/dx, use the chain rule. In this case, du/dx = (1/(2x)) * (1/x) = 1/(2x^2).
Step 3: Solve for dx in terms of du.
Rearranging the equation du/dx = 1/(2x^2), we get dx = 2x^2 du.
Step 4: Rewrite the integral.
Substituting the new variables, we have:
∫(5dx)/(xln(2x)) = ∫(5 * 2x^2 du)/(x * u)
Step 5: Simplify the integrand.
In this step, we simplify the above equation:
∫(10x du)/(u) = 10∫(x du)/(u)
Step 6: Evaluate the integral.
Integrating, we have:
10∫(x du)/(u) = 10 ln|x| + C
Therefore, the indefinite integral of (5dx)/(xln(2x)) is 10 ln|x| + C, where C represents the constant of integration.