$$\int_a^a f(x)\,dx$$
$$0$$
$$\int_a^b k\,dx$$
$$k(b-a)$$
$$\int_a^b k f(x)\,dx$$
$$k\int_a^b f(x)\,dx$$
$$\int_a^b f(x)\pm g(x)\,dx$$
$$\int_a^b f(x)\,dx\pm\int_a^b g(x)\,dx$$
$$\int_a^{-a} \text{odd}(x)\,dx$$
$$0$$
$$\int_{-a}^a \text{even}(x)\,dx$$
$$2\int_0^a \text{even}(x)\,dx$$
$$\int_a^b f(x)\,dx$$
$$-\int_b^a f(x)\,dx$$
$$\int_a^b f(x)\,dx$$
$$\int_a^c f(x)\,dx+\int_c^b f(x)\,dx$$
$$\int x^n\,dx\,,n\ne1$$
$$\frac{x^{n+1}}{n+1}+C$$
$$\int \frac{1}{x}\,dx$$
$$\ln{|x|}+C$$
$$\int \sin{x}\,dx$$
$$-\cos{x}+C$$
$$\int \cos{x}\,dx$$
$$\sin{x}+C$$
$$\int \sec^2{x}\,dx$$
$$\tan{x}+C$$
$$\int \csc^2{x}\,dx$$
$$-\cot{x}+C$$
$$\int \sec{x}\tan{x}\,dx$$
$$\sec{x}+C$$
$$\int \csc{x}\cot{x}\,dx$$
$$-\csc{x}+C$$
$$\int e^{x}\,dx$$
$$e^{x}+C$$
$$\int \frac{1}{\sqrt{1-x^2}}\,dx$$
$$\sin^{-1}{x}+C$$
$$\int \frac{1}{1+x^2}\,dx$$
$$\tan^{-1}{x}+C$$
$$\int f'(mx+b)\,dx$$
$$\frac{f(mx+b)}{m}+C$$
IBP
$$\int uv’\,dx=uv-\int u’v\,dx$$
IBP
$$\int_a^b uv’\,dx=uv\rvert_a^b-\int_a^b u’v\,dx$$
$$\text{new}=\text{old}+\text{change}$$
$$f(b)=f(a)+\int_a^b f'(x)\,dx$$
FTC Part I
$$\frac{d}{dx}\int_a^x f(t)\,dt=f(x)$$
FTC Part II
$$\int_a^b f(x)\,dx=F(b)-F(a),$$$$F'(x)=f(x)$$