Integrals (Advanced)

Harder items at competition-adjacent depth.

Phase 2 · AP Calculus

Integrals: core idea

u-substitution.

Evaluate $\int_0^1 x\ln x\,dx$ .

Hint: Parts; limit at 0. Answer: -1/4.

1. Evaluate $\int_0^{\infty}e^{-x}\,dx$ .
2. Volume rotating $y=x$ on $[0,1]$ about $x$ -axis?
3. Evaluate $\int_0^1 \dfrac{x}{1+x^2}\,dx$ .
4. Evaluate $\int_0^1 \dfrac{dx}{\sqrt{1-x^2}}$ .
5. Arc length of $y=x$ on $[0,1]$ ?
6. Evaluate $\int x^2 e^x\,dx$ at bounds $0$ to $1$ : $F(1)-F(0)$ with $F=e^x(x^2-2x+2)$ ?
7. Evaluate $\int_0^{\pi/2}\sin^2 x\cos x\,dx$ .
8. Evaluate $\int_1^2 \dfrac{2x}{x^2+1}\,dx$ as $\ln(5)/k$ . Find $k$ .
9. Shell method: rotate $y=\sqrt{x}$ , $0\le x\le1$ about $y$ -axis: volume $/\pi$ ?