Question

Find: \( \int 2x^3 e^{x^2} \,dx \).

Hint:

For integrals involving \( x e^{x^2} \), try substitution \( u = x^2 \) and use integration by parts if necessary.

Answer 1

Step 1: Identify substitution.
The given integral is: \[ I = \int 2x^3 e^{x^2} \,dx. \] We use the substitution: \[ u = x^2 \quad \Rightarrow \quad du = 2x \,dx. \] Step 2: Transform the integral.
Rewriting in terms of \( u \): \[ \int 2x^3 e^{x^2} \,dx = \int x^2 \cdot 2x e^{x^2} \,dx. \] Since \( 2x \,dx = du \), we substitute: \[ I = \int x^2 e^u \,du. \] Since \( x^2 = u \), we get: \[ I = \int u e^u \,du. \] Step 3: Integration by parts.
Using integration by parts, where: \[ \int u v' \,du = u v - \int v u' \,du, \] let: \[ u = u, \quad dv = e^u \,du. \] Then: \[ du = du, \quad v = e^u. \] Applying integration by parts: \[ I = u e^u - \int e^u \,du. \] Since \( \int e^u \,du = e^u \), we get: \[ I = u e^u - e^u + C. \] Step 4: Substituting back \( u = x^2 \).
\[ I = x^2 e^{x^2} - e^{x^2} + C. \] Final Answer: \[ \int 2x^3 e^{x^2} \,dx = (x^2 - 1) e^{x^2} + C. \]