Integration by parts: definite integrals | AP Calculus BC | Khan Academy

Introduction

This tutorial will guide you through the process of solving definite integrals using integration by parts, a key technique in calculus. Understanding this method is essential for AP Calculus BC students, as it helps simplify complex integrals into more manageable forms, ultimately leading to accurate evaluations.

Step 1: Identify the Integral

• Start with the definite integral you need to evaluate. For example, consider the integral of the form (\int_a^b u , dv).
• Choose functions (u) and (dv) from your integral. It's crucial to select (u) as the function that simplifies upon differentiation, while (dv) should be easily integrable.

Step 2: Differentiate and Integrate

• Differentiate (u) to find (du):
  • (du = \frac{d}{dx}(u) , dx)
• Integrate (dv) to find (v):
  • (v = \int dv)

Step 3: Apply the Integration by Parts Formula

• Utilize the integration by parts formula:
  • (\int u , dv = uv - \int v , du)
• Substitute (u), (v), and (du) into the formula.

Step 4: Evaluate the Antiderivative

• Evaluate the expression (uv) at the boundaries (a) and (b):
  • Calculate (uv|_a^b = uv(b) - uv(a))
• Substitute back into the formula:
  • (\int_a^b u , dv = uv|_a^b - \int_a^b v , du)

Step 5: Calculate the Remaining Integral

• Solve the remaining integral (\int v , du).
• If this integral can also be evaluated, proceed to find its value.

Step 6: Combine Results

• Combine your results from Step 4 and Step 5 to get the final value of the definite integral:
  • Final result = (uv|_a^b - \text{(value of the remaining integral)})

Practical Tips

• When choosing (u) and (dv), consider using the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to determine the best choice.
• Be careful with the limits of integration; they apply to every term in your final expression.
• If the integral involves products of functions, you may need to apply integration by parts multiple times.

Common Pitfalls

• Forgetting to apply the limits of integration correctly after finding the antiderivative.
• Choosing (u) and (dv) poorly, leading to more complex integrals instead of simplifying the problem.
• Neglecting to simplify expressions after substituting back into the integration by parts formula.

Conclusion

Integration by parts is a powerful method for evaluating definite integrals. By carefully choosing your functions and systematically applying the formula, you can simplify complex integrals into solvable forms. Practice this technique with various integrals to build confidence and mastery. For further learning, consider exploring additional resources or practice problems on Khan Academy.