I don't know how to do the integral of e^(lnx^2)dx and the integral of (sin sqrtx)/(sqrtx) dx
1. ∫e^(lnx^2)dx
use the identity e^(ln(y)) = y to simplify the expression.
2. try the substitution u=sqrt(x).
Thanks for help on 1.
on 2. if i do u=sqrt(x) my du is 1/2x^(-1/2) and that means my du is in the denominator. So it would read 2integral of sin(u)/du
Nope, not on two.
You cant solve it that way easily.
This is difficult. Brake the sin function into its series equivalent, and integrate the series.
http://reference.wolfram.com/mathematica/ref/SinIntegral.html
forget that last answer. I am tired.
For 2, almost, but not quite!
Start with:
u=√x
du = (1/2)dx/√x
dx/√x = 2du
so
∫sin(√x) dx/√x
=∫sin(u)*2du
=-2cos(u)
=-2cos(√x)
To solve the integral of e^(lnx^2), we can simplify the expression first.
Step 1: Simplify the expression
Using the property of logarithms, ln(x^a) = a * ln(x), we can rewrite e^(lnx^2) as x^2.
Step 2: Integrate x^2
Now that we have simplified the expression, we need to integrate x^2.
The integral of x^n dx, where n is any constant except -1, is (x^(n+1)) / (n+1) + C, where C is the constant of integration.
Applying this formula, the integral of x^2 dx is (1/3) * x^3 + C.
Therefore, the integral of e^(lnx^2)dx is equal to (1/3) * x^3 + C, where C is the constant of integration.
Next, let's solve the integral of (sin(sqrtx)) / sqrtx dx.
Step 1: Substitute variables
Let's substitute sqrtx with a new variable u. This would mean u^2 = x, which implies 2u du = dx. Rearranging, we find dx = 2u du.
Step 2: Rewrite the integral
Rewriting the integral with respect to the variable u, we get ∫ (sin u) / u du.
Step 3: Solve the integral
The integral of (sin u) / u can be solved using a special function called the sine integral (Si). The sine integral of u is defined as ∫ (sin t) / t dt.
Therefore, the integral of (sin u) / u is equal to Si(u) + C, where C is the constant of integration.
Step 4: Substitute back the variable
Now, substituting back u = sqrtx, we have Si(sqrtx) + C as our final result.
Hence, the integral of (sin(sqrtx)) / sqrtx dx is equal to Si(sqrtx) + C, where Si(x) is the sine integral function and C is the constant of integration.