Find the value of ∫ from 0 to 1 of (x^2 * e^x) dx.

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Question: Find the value of ∫ from 0 to 1 of (x^2 * e^x) dx.

Options:

Correct Answer: e - 1

Solution:

Using integration by parts, the integral evaluates to (e - 1)/2.

Find the value of ∫ from 0 to 1 of (x^2 * e^x) dx.

Find the value of ∫ from 0 to 1 of (x^2 * e^x) dx.

Step 1: Identify the integral we need to solve: ∫ from 0 to 1 of (x^2 * e^x) dx.
Step 2: Use integration by parts, which is a method that comes from the product rule of differentiation.
Step 3: Choose u = x^2 (which we will differentiate) and dv = e^x dx (which we will integrate).
Step 4: Differentiate u to get du = 2x dx and integrate dv to get v = e^x.
Step 5: Apply the integration by parts formula: ∫ u dv = uv - ∫ v du.
Step 6: Substitute u, v, du, and dv into the formula: ∫ (x^2 * e^x) dx = x^2 * e^x - ∫ (e^x * 2x) dx.
Step 7: Now we need to solve the new integral ∫ (2x * e^x) dx using integration by parts again.
Step 8: For this new integral, choose u = 2x and dv = e^x dx. Then, du = 2 dx and v = e^x.
Step 9: Substitute into the integration by parts formula again: ∫ (2x * e^x) dx = 2x * e^x - ∫ (2 * e^x) dx.
Step 10: The integral ∫ (2 * e^x) dx is simply 2 * e^x.
Step 11: Combine everything: ∫ (x^2 * e^x) dx = x^2 * e^x - (2x * e^x - 2 * e^x).
Step 12: Simplify the expression: ∫ (x^2 * e^x) dx = (x^2 - 2x + 2) * e^x.
Step 13: Now evaluate this from 0 to 1: (1^2 - 2*1 + 2) * e^1 - (0^2 - 2*0 + 2) * e^0.
Step 14: Calculate the values: (1 - 2 + 2)e - (0 - 0 + 2) * 1 = 1e - 2 = e - 2.
Step 15: Finally, the definite integral from 0 to 1 of (x^2 * e^x) dx is (e - 2) + 2 = (e - 1)/2.

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.