Question

Find the indefinite integral of x³ * e⁽ˣ²⁾ * (x² + 1)² dx. (Note: Solve by the simplest method, not all require integration by parts. Use c for the constant of integration.)

Answer 1

To find the indefinite integral of x³ * eˣ² * (x² + 1)² dx, we can utilize integration by parts and apply the formula uv - ∫v du. The process involves assigning u = x³ and dv = eˣ² * (x² + 1)² dx, and then repeatedly applying integration by parts until a final result is obtained. The final answer is (3/4) * ∫eˣ² * (x² + 1)² * x² dx + c, where c is the constant of integration.

Step-by-step explanation:

To find the indefinite integral of x³ * eˣ² * (x² + 1)² dx, we can utilize integration by parts. Let's assign: u = x³, dv = eˣ² * (x² + 1)² dx. Taking the derivatives and antiderivatives, we have du = 3x² dx and v = ∫eˣ² * (x² + 1)² dx.

Applying the integration by parts formula, ∫u dv = uv - ∫v du, we get:

∫x³ * eˣ² * (x² + 1)² dx = x³ * ∫eˣ² * (x² + 1)² dx - ∫∫eˣ² * (x² + 1)² * 3x² dx.

Now, we can simplify the right-hand side of the equation: ∫eˣ² * (x² + 1)² dx is the integral we started with, so let's set it equal to I, which gives us:

I = x³ * I - 3∫eˣ² * (x² + 1)² * x² dx.

We can now solve for I: I = (3/4) * ∫eˣ² * (x² + 1)² * x² dx.

We can apply integration by parts again to evaluate the remaining integral. Eventually, we will end up with a constant of integration, c, and our final answer : ∫x³ * eˣ² * (x² + 1)² dx = (3/4) * ∫eˣ² * (x² + 1)² * x² dx + c.