# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- --- # **Key Concepts** This exercise focuses on the following concepts: - Integration by parts formula $\int u\,dv = uv - \int v\,du$ - LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) for selecting $u$ and $dv$ - Repeated integration by parts for polynomial factors (tabular integration) - Cyclic integrals where integration by parts returns the original form (e.g., $e^x \cos x$) - Integration of inverse trigonometric functions via integration by parts - Combining substitution with integration by parts - Reduction formulas for powers of logarithmic functions --- # **Important Formulas** Below are the key formulas used in this exercise: **Integration by Parts:** $$\int u \, dv = uv - \int v \, du$$ **Cyclic Integral Pattern** (for $\int e^{ax}\cos(bx)\,dx$ or $\int e^{ax}\sin(bx)\,dx$): $$\int e^{ax}\cos(bx)\,dx = \frac{e^{ax}}{a^2+b^2}(a\cos(bx) + b\sin(bx)) + C$$ **Standard Logarithmic Integral:** $$\int \ln x \, dx = x\ln x - x + C$$ --- # **Summary** This exercise develops proficiency in integration by parts across diverse function types. The key strategy involves selecting $u$ using the LIATE priority rule: logarithmic and inverse trig functions are typically chosen as $u$, while exponentials and trig functions are assigned to $dv$. For integrals with polynomial factors (Questions 4, 9, 13, 14, 16, 17), repeated integration by parts (tabular method) is efficient until the polynomial reduces to a constant. Questions 3 and 15 require combining substitution with integration by parts. Question 6 demonstrates the cyclic pattern where integration by parts twice yields an equation solvable for the original integral. Inverse trig integrals (Questions 7, 11, 12, 14) are handled by setting $u$ as the inverse function and $dv = dx$.