How do I evaluate the integral of x³ cos(x²) dx using substitution and integration by parts?

To evaluate the integral ∫ x³ cos(x²) dx, we will first use substitution and then apply integration by parts.

We start with a substitution that simplifies our integral. Let’s set:

u = x²

Then, the differential du is:

du = 2x dx or dx = rac{du}{2x}

Now, we can express x³ in terms of u. Since x = ext{sqrt}(u), we have:

x³ = (u^{1/2})^3 = u^{3/2}

This gives us:

dx = rac{du}{2 ext{sqrt}(u)}

Now let’s rewrite the original integral in terms of u:

∫ x³ cos(x²) dx = ∫ u^{3/2} cos(u) rac{du}{2 ext{sqrt}(u)} = rac{1}{2} ∫ u ext{cos}(u) du

Now we can use integration by parts to evaluate ∫ u ext{cos}(u) du. According to the integration by parts formula:

∫ v rac{du}{dx} dx = u v – ∫ v rac{du}{dx} dx

Let’s choose:

Applying the integration by parts formula, we have:

∫ u ext{cos}(u) du = u ext{sin}(u) – ∫ ext{sin}(u) du

Now we find ∫ ext{sin}(u) du, which evaluates to:

– ext{cos}(u)

So, plugging this back into our equation, we get:

∫ u ext{cos}(u) du = u ext{sin}(u) + ext{cos}(u) + C

Now substituting back the value of u = x²:

= x² ext{sin}(x²) + ext{cos}(x²) + C

Returning to our original integral:

∫ x³ cos(x²) dx = rac{1}{2} igg(x² ext{sin}(x²) + ext{cos}(x²) + C igg)

Therefore, the final answer is:

∫ x³ cos(x²) dx = rac{1}{2} igg(x² ext{sin}(x²) + ext{cos}(x²)igg) + C