Evaluate the integral from

[sqrt(pi/2), sqrt(pi)] of x^3*cos(x^2) by first making a substitution and then using integration by parts.

I let u = x^2 and du= 2x dx but then it doesn't equal that in the equation?

∫x^3 cos(x^2) dx

If you let u=x^2, you have du = 2x dx and the integral is
1/2 ∫[π/2,π] u cos(u) du

Now you can tackle that using integration by parts.

To evaluate the integral ∫[sqrt(pi/2), sqrt(pi)] x^3*cos(x^2) dx, you correctly made the substitution u = x^2. However, it seems you made a mistake in calculating the differential du.

Let's correctly calculate du and proceed with the solution.

Given: u = x^2, we can differentiate both sides of this equation with respect to x to find du/dx:
du/dx = 2x.

Now, we need to express dx in terms of du to rewrite the integral in terms of u. To do this, we can solve the derivative expression for dx:
dx = (1 / (2x)) du.

Substituting the values of x and dx into the original integral, we get:
∫[sqrt(pi/2), sqrt(pi)] x^3*cos(x^2) dx = ∫[sqrt(pi/2), sqrt(pi)] (x^3*cos(x^2)) * ((1 / (2x)) du).

Now, the x term in the integral can be canceled out, and we are left with:
∫[sqrt(pi/2), sqrt(pi)] (1/2) * (x^2*cos(x^2)) du.

The next step is to apply integration by parts. Integration by parts is given by the formula:
∫ u * dv = uv - ∫ v * du.

To apply this rule, we need to identify u and dv. Let's choose u = x^2 and dv = (1/2) * cos(x^2) du.

Now, we can calculate du by differentiating u:
du = 2x dx.

To find v, we integrate dv:
v = (1/2) * ∫ cos(x^2) du.

At this point, finding the integral of cos(x^2) with respect to x is not straightforward. The integral does not have a simple, elementary form and does not have standard antiderivatives like sin(x), cos(x), or e^x. Thus, it does not lead to a closed-form solution.

Therefore, the final result will be expressed in terms of the integral itself:
∫ cos(x^2) dx.

To obtain a numerical approximation of the integral, you can use numerical integration methods like Simpson's rule or the trapezoidal rule. These methods approximate the integral by breaking the integration interval into smaller segments and approximating the function within each segment.

In summary, after correctly making the substitution u = x^2, you can rewrite the integral in terms of u. However, the resulting integral cannot be evaluated in closed form, and numerical methods should be used for approximation.