1.Evaluate $\int \frac{x^2 - 1}{x^4 + 1} dx$.
2.Evaluate $\int \frac{x^2 + 1}{x^4 + x^2 + 1} dx$.
3.Find $\int (\sqrt{\tan x} + \sqrt{\cot x}) dx$.
4.Evaluate $\int \frac{dx}{x^4+1}$.
5.Find $\int \frac{dx}{x^2(x^4+1)^{3/4}}$.
6.Evaluate $\int \frac{\sin x}{\sin(x-a)\sin(x-b)} dx$.
7.Find $\int e^x \left( \frac{1-\sin x}{1-\cos x} \right) dx$.
8.Evaluate $\int \frac{x^2+1}{(x\sin x + \cos x)^2} dx$.
9.Find $\int \frac{dx}{\sin^4 x + \cos^4 x}$.
10.Evaluate $\int \frac{x^3 - 1}{x^3 + x} dx$.
11.Find $\int \frac{dx}{(x-1)^{3/4}(x+2)^{5/4}}$.
12.Evaluate $\int \frac{\sin x + \cos x}{9 + 16\sin 2x} dx$.
13.Find $\int \sin^{-1}\sqrt{\frac{x}{a+x}} dx$.
14.Evaluate $\int \frac{e^x(x^2+1)}{(x+1)^2} dx$.
15.Find $\int \frac{\sqrt{x^2+1}}{x^4} dx$.
16.Evaluate $\int \frac{x e^x}{(x+1)^2} dx$.
17.Find $\int_0^{\pi/2} \ln(\sin x) dx$.
18.Evaluate $\int_0^\pi \frac{x \tan x}{\sec x + \tan x} dx$.
19.Find $\int_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} dx$.
20.Evaluate $\int_0^\pi x \ln(\sin x) dx$.
21.Find $\int_0^1 \frac{\ln(1+x)}{1+x^2} dx$.
22.Evaluate $\int_0^{\pi/4} \ln(1+\tan x) dx$.
23.Find the value of $\int_0^{\infty} \frac{\ln x}{1+x^2} dx$.
24.Evaluate $\lim_{n \to \infty} \sum_{r=1}^n \frac{n}{n^2+r^2}$.
25.Evaluate $\lim_{n \to \infty} \left( \frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{3n} \right)$.
26.Let $I_n = \int_0^{\pi/4} \tan^n x dx$. Prove that $I_n + I_{n-2} = \frac{1}{n-1}$. Hence evaluate $I_5 + I_3$.
27.Evaluate $\int_{-1}^1 \frac{x^3 + |x| + 1}{x^2 + 2|x| + 1} dx$.
28.Find $\int_0^{100\pi} \sqrt{1-\cos 2x} dx$.
29.Evaluate $\int_0^2 \{x\} dx$, where $\{x\}$ denotes the fractional part of $x$.
30.Evaluate $\int_{-1}^1 [x^2 + \{x\}] dx$, where $[x]$ is the greatest integer function and $\{x\}$ is the fractional part.
31.Evaluate $\int_0^a f(x)g(x) dx$ if it is given that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$.
32.Find $\int_0^{\pi/2} \frac{1}{1+\sqrt{\tan x}} dx$.
33.Evaluate $\int_0^\pi \frac{e^{\cos x}}{e^{\cos x} + e^{-\cos x}} dx$.
34.If $f(x)$ is an even function, show that $\int_{-a}^a \frac{f(x)}{1+e^x} dx = \int_0^a f(x) dx$.
35.Evaluate $\int_{1/e}^e |\ln x| dx$.
36.Find $\int_0^1 x \sqrt{\frac{1-x^2}{1+x^2}} dx$.
37.Evaluate $\int_{-\pi/2}^{\pi/2} (\sin|x| + \cos|x|) dx$.
38.Evaluate $\int_0^1 \cot^{-1}(1-x+x^2) dx$.
39.If $F(x) = \int_{x^2}^{x^3} \frac{1}{\log t} dt$, find $F'(x)$ using Leibniz's Rule.
40.Evaluate $\int_0^{\pi^2/4} \sin\sqrt{x} dx$.
41.Evaluate the limit: $\lim_{x \to 0} \frac{\int_0^{x^2} \cos(t^2) dt}{x \sin x}$.
42.Find $\int_0^\pi \frac{x}{a^2\cos^2x + b^2\sin^2x} dx$ (where $a, b > 0$).
43.Evaluate $\int_0^{\pi/2} \sin 2x \ln(\tan x) dx$.
44.Find $\int_{-1}^1 \frac{\sin x - x^2}{3-|x|} dx$.
45.Evaluate $\int_0^1 \frac{x^4(1-x)^4}{1+x^2} dx$.
46.Evaluate $\int_{-2}^2 \min(x-[x], -x-[-x]) dx$.
47.Let $f(x)$ be continuous and $\int_0^1 f(x)dx = 2$. Evaluate $\int_0^1 f(1-x)dx$.
48.Evaluate $\int_{0}^{2\pi} \frac{x \sin^{2n} x}{\sin^{2n} x + \cos^{2n} x} dx$.
49.Find $\int_{-1}^1 \ln\left( x + \sqrt{x^2+1} \right) dx$.
50.Find $\int \frac{\cos x - \sin x}{\sqrt{8 - \sin 2x}} dx$.
51.Evaluate $\int \frac{\sin 2x}{\sin^4 x + \cos^4 x} dx$.
52.Find $\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} dx$.
53.Evaluate $\int \frac{x^2-1}{x\sqrt{x^4+3x^2+1}} dx$.
54.Evaluate $\int \frac{1}{\cos(x-a)\cos(x-b)} dx$.
55.Find $\int \cos(2\theta) \ln\left(\frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta}\right) d\theta$.
56.Evaluate $\int \frac{dx}{(x+1)\sqrt{x^2-1}}$.
57.If $I = \int_0^1 \frac{\sin^{-1}x}{x} dx$, show its connection to the series $\sum_{n=1}^\infty \frac{1}{n^2}$. (Just set up the Maclaurin expansion).
58.Evaluate $\int_0^{\pi/2} \frac{x}{\sin x} dx$.
59.Find $\int_0^1 \frac{\log(1+x)}{x} dx$.
60.Evaluate $\int \frac{x^2}{(x\sin x + \cos x)^2} dx$.
