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: Substitution Identification.
The integral provided is: \[ I = \int 2x^3 e^{x^2} \,dx. \] The substitution employed is: \[ u = x^2 \quad \Rightarrow \quad du = 2x \,dx. \] Step 2: Integral Transformation.
Expressing the integral in terms of \( u \): \[ \int 2x^3 e^{x^2} \,dx = \int x^2 \cdot 2x e^{x^2} \,dx. \] Substituting \( 2x \,dx = du \): \[ I = \int x^2 e^u \,du. \] Substituting \( x^2 = u \): \[ I = \int u e^u \,du. \] Step 3: Integration by Parts.
Applying the integration by parts formula, \( \int u v' \,du = u v - \int v u' \,du \), with: \[ u = u, \quad dv = e^u \,du. \] This yields: \[ du = du, \quad v = e^u. \] Performing the integration by parts: \[ I = u e^u - \int e^u \,du. \] Since \( \int e^u \,du = e^u \): \[ I = u e^u - e^u + C. \] Step 4: Back-substitution.
Substituting \( u = x^2 \) back into the expression: \[ I = x^2 e^{x^2} - e^{x^2} + C. \] Final Result: \[ \int 2x^3 e^{x^2} \,dx = (x^2 - 1) e^{x^2} + C. \]