1 Integration by parts / 4 pts
(a) Find \( \int x^2\ln x\,dx \). (b) Evaluate \( \int_0^{\pi}x\sin x\,dx \).
2 Trigonometric integrals / 3 pts
(a) Find \( \int\cos^2x\,dx \). (b) Find \( \int\sin^2x\cos^3x\,dx \).
3 Trigonometric substitution / 4 pts
(a) Find \( \int\dfrac{dx}{x^2\sqrt{4-x^2}} \) for \(0\lt x\lt2\). (b) Evaluate \( \int_0^3\dfrac{dx}{x^2+9} \).
4 Partial fractions / 4 pts
Let \( f(x)=\dfrac{x+5}{x^2-x-6} \). (a) Decompose \(f\). (b) Find an antiderivative. (c) Evaluate \( \int_4^6f(x)\,dx \) to three decimals (calculator allowed).
5 Improper integrals / 3 pts
(a) Evaluate \( \int_1^{\infty}\dfrac{\ln x}{x^2}dx \). (b) Evaluate \( \int_0^1x^{-1/3}dx \).
6 Numerical integration / 2 pts
Use the trapezoid rule with \(n=4\) to approximate \( \int_0^2x^2dx \) and find the error compared with the exact value.
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