Calculate ∫ from 0 to 1 of (x^2 * e^x) dx.

Calculate ∫ from 0 to 1 of (x^2 * e^x) dx.

Step 1: Identify the integral we need to calculate: ∫ from 0 to 1 of (x^2 * e^x) dx.
Step 2: Use integration by parts, which is based on the formula ∫ u dv = uv - ∫ v du.
Step 3: Choose u = x^2 and dv = e^x dx. Then, calculate du and v.
Step 4: Differentiate u: du = 2x dx.
Step 5: Integrate dv: v = e^x.
Step 6: Apply the integration by parts formula: ∫ x^2 e^x dx = x^2 e^x - ∫ e^x (2x) dx.
Step 7: Now we need to calculate the new integral ∫ 2x e^x dx using integration by parts again.
Step 8: Choose u = 2x and dv = e^x dx. Then, calculate du and v.
Step 9: Differentiate u: du = 2 dx.
Step 10: Integrate dv: v = e^x.
Step 11: Apply the integration by parts formula again: ∫ 2x e^x dx = 2x e^x - ∫ e^x (2) dx.
Step 12: The integral ∫ e^x (2) dx = 2 e^x.
Step 13: Combine everything: ∫ x^2 e^x dx = x^2 e^x - (2x e^x - 2 e^x).
Step 14: Simplify the expression: ∫ x^2 e^x dx = x^2 e^x - 2x e^x + 2 e^x.
Step 15: Now evaluate from 0 to 1: [1^2 e^1 - 2(1)e^1 + 2e^1] - [0^2 e^0 - 2(0)e^0 + 2e^0].
Step 16: Calculate the values: (1/e - 2/e + 2/e) - (0 - 0 + 2) = (1/e) - 2.
Step 17: The final result is (2/e - 1/e) = 1/e.

Integration by Parts – A technique used to integrate products of functions, based on the formula ∫u dv = uv - ∫v du.
Definite Integrals – Calculating the area under a curve between specified limits, in this case from 0 to 1.
Exponential Functions – Functions of the form e^x, which are important in calculus and often appear in integration problems.