A few definite integral practice problems:

1. $${\int_{0}^{\pi} 3\cos^2(x)\sin(x)\,dx}$$ (ANSWER: $2$)
2. $${\int_{1}^{2} x\ln(x)\,dx}$$ (ANSWER: $\ln(4)-\frac{3}{4}$)
3. $${\int_{0}^{\pi/2} x^2\sin(2x)\,dx}$$ (ANSWER: $\frac{\pi^2-4}{8}$)
4. $${\int_{-1}^{1} \frac{5x}{(4+x^2)^2}\,dx}$$ (ANSWER: $0$)
5. $${\int_{0}^{\pi/6} (1-\cos(3x)) \sin(3x)\,dx}$$ (ANSWER: $\frac{1}{6}$)
6. $${\int_{1}^{0} \sqrt{x^5+2x}(5x^4+2)\,dx}$$ (ANSWER: $-2\sqrt{3}$)