Evaluate using u-substitution:

Integral of: 4x(tan(x^2))dx

Integral of: (1/(sqrt(x)*x^(sqrt(x))))dx

Integral of: (cos(lnx)/x)dx

Integral of: (cos(lnx)/x)dx

let u = ln x
du = dx/x
Your integral is therefore that of
cos u du, which is sin u = sin(ln x)

Try the others yourself. In the first one, I would let x^2 = u

Integral 3x/x2(x-2)dx

To evaluate the given integrals using u-substitution, follow these steps:

1. Identify the u-term: Look for a part of the integrand that can be denoted by a single variable u. Choose this part as the u-term.

2. Calculate du: Find the derivative of the u-term with respect to x. This will give you du/dx.

3. Rewrite the integral in terms of u and du: Replace the u-term with u and dx with du in the integral.

4. Evaluate the integral with respect to u: After rewriting the integral in terms of u and du, evaluate it as a simpler integral with respect to u.

5. Convert back to the original variable: Replace u in the result with the original variable x to obtain the final solution.

Let's apply these steps to the given integrals:

1. Integral of: 4x(tan(x^2))dx

Let u = x^2
du/dx = 2x
dx = du / (2x)

Rewriting the integral:
∫4x(tan(x^2))dx = ∫4(u)(tan(u))(du / (2x))

Simplifying the integral:
= ∫2(tan(u))du

Now we can evaluate the integral with respect to u:
∫2(tan(u))du = 2∫tan(u)du = 2 * -ln|cos(u)| + C, where C is the constant of integration.

Converting back to the original variable:
= -2ln|cos(u)| + C
= -2ln|cos(x^2)| + C

2. Integral of: (1/(sqrt(x)*x^(sqrt(x))))dx

Let u = x^sqrt(x)
du/dx = sqrt(x)*x^(sqrt(x) - 1)
dx = (du / (sqrt(x)*x^(sqrt(x) - 1))

Rewriting the integral:
∫(1/(sqrt(x)*x^(sqrt(x))))dx = ∫(1/u) * (du / (sqrt(x)*x^(sqrt(x) - 1)))

Simplifying the integral:
= ∫(1/u) * (du / (sqrt(x)*x^(sqrt(x) - 1)))
= ∫(1/u) * (1/(sqrt(x)*x^(1 - sqrt(x)))) du

Now we can evaluate the integral with respect to u:
∫(1/u) * (1/(sqrt(x)*x^(1 - sqrt(x)))) du = ∫(1/u) * (1/(sqrt(u)*u^(1 - sqrt(u)))) du

Let's simplify it further:
= ∫(u^sqrt(u)*sqrt(u)^(-1))/(u^1*u^(-sqrt(u)))) du
= ∫(1/(u^(1 - sqrt(u)))) du

Now we can evaluate the integral with respect to u:
∫(1/(u^(1 - sqrt(u)))) du = -((u^(-sqrt(u)))/(sqrt(u))) + C

Converting back to the original variable:
= -((x^(-sqrt(x^sqrt(x))))/(sqrt(x^sqrt(x)))) + C

3. Integral of: (cos(lnx)/x)dx

Let u = ln(x)
du/dx = 1/x
dx = x * du

Rewriting the integral:
∫(cos(lnx)/x)dx = ∫(cos(u) * (x * du))

Simplifying the integral:
= ∫cos(u) * x du

Now we can evaluate the integral with respect to u:
∫cos(u) * x du = x * ∫cos(u) du = x * sin(u) + C, where C is the constant of integration.

Converting back to the original variable:
= x * sin(ln(x)) + C

Note: The antiderivative is not always expressible in elementary functions, and in such cases, the integral may not have a simple closed-form expression.