Integration by Parts Application

Incorrect

Evaluate the following indefinite integral:

∫ x² sin(x) dx

cancel YOUR ANSWER

-x² cos(x) + 2x sin(x) + C

check_circle CORRECT ANSWER

-x² cos(x) + 2x sin(x) + 2 cos(x) + C

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The Explanation

This problem requires two iterations of integration by parts. Let's break down the logic step-by-step:

First Pass: Let u = x² (algebraic) and dv = sin(x)dx (trigonometric). Following the ILATE rule, we find du = 2xdx and v = -cos(x). The formula uv - ∫vdu gives us -x²cos(x) + ∫2x cos(x)dx.
Second Pass: We must integrate ∫2x cos(x)dx. Again, let u = 2x and dv = cos(x)dx. This yields du = 2dx and v = sin(x). Applying the formula again gives 2x sin(x) - ∫2sin(x)dx.
Final result: Integrating the final term gives +2cos(x). Combine all parts for the full solution.

u

dv

Visualizing the Tabular Method for Repeated Integration

flag Why you might have missed this

Conceptual misunderstanding: You successfully performed the first iteration but failed to carry the second integration to completion, missing the final trigonometric term constant.

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Advanced Integration Patterns arrow_forward

05

Fundamental Theorem of Calculus

Correct

Find the derivative of the function:

G(x) = ∫₀^x³ sqrt(1 + t²) dt

Great job! You correctly applied the Chain Rule alongside the Fundamental Theorem. Most students forget to multiply by the derivative of the upper limit (3x²).