Fundamental Theorem of Calculus
$$\int_a^b f(x)\,dx = F(b)-F(a)\quad \text{where }F'(x)=f(x)$$
Linearity of Indefinite Integrals
$$\int \bigl(f(x)\pm g(x)\bigr)\,dx=\int f(x)\,dx \pm \int g(x)\,dx$$
$$\int kf(x)\,dx = k\int f(x)\,dx$$
$$\int c\,dx = cx + C$$
Power and Logarithmic Integrals
$$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\quad (n\ne -1)$$
$$\int \frac{dx}{x}=\ln|x|+C$$
Exponential Integrals
$$\int e^x\,dx=e^x+C$$
$$\int a^x\,dx=\frac{a^x}{\ln a}+C\quad (a>0,\;a\ne 1)$$
Definite Integral Properties
$$\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$$
$$\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx$$
$$\int_a^b kf(x)\,dx = k\int_a^b f(x)\,dx$$
$$\int_a^b \bigl(f(x)\pm g(x)\bigr)\,dx=\int_a^b f(x)\,dx \pm \int_a^b g(x)\,dx$$
Substitution Rule
$$u=g(x),\;du=g'(x)\,dx\quad\Rightarrow\quad \int f(g(x))\,g'(x)\,dx=\int f(u)\,du$$
$$\int_a^b f(g(x))\,g'(x)\,dx=\int_{u(a)}^{u(b)} f(u)\,du$$
Integration by Parts
$$\int u\,dv = uv - \int v\,du$$
$$\int_a^b u\,dv=\Big[uv\Big]_a^b-\int_a^b v\,du$$
Product-to-Sum Identities
$$2\sin A\sin B=\cos(A-B)-\cos(A+B)$$
$$2\cos A\cos B=\cos(A-B)+\cos(A+B)$$
$$2\sin A\cos B=\sin(A-B)+\sin(A+B)$$
Power-Reduction and Pythagorean Identities
$$\sin^2 x=\frac{1-\cos 2x}{2}$$
$$\cos^2 x=\frac{1+\cos 2x}{2}$$
$$1+\tan^2 x=\sec^2 x$$
$$1+\cot^2 x=\csc^2 x$$
Basic Trigonometric Integrals
$$\int \sin u\,du = -\cos u + C$$
$$\int \cos u\,du = \sin u + C$$
$$\int \sec^2 u\,du = \tan u + C$$
$$\int \csc^2 u\,du = -\cot u + C$$
$$\int \sec u\,\tan u\,du = \sec u + C$$
$$\int \csc u\,\cot u\,du = -\csc u + C$$
$$\int \tan u\,du = \ln|\sec u| + C$$
$$\int \cot u\,du = \ln|\sin u| + C$$
$$\int \sec u\,du = \ln|\sec u+\tan u| + C$$
$$\int \csc u\,du = \ln|\csc u-\cot u| + C$$
Inverse Trigonometric Forms
$$\int \frac{du}{a^2+u^2}=\frac{1}{a}\tan^{-1}\!\left(\frac{u}{a}\right)+C$$
$$\int \frac{du}{\sqrt{a^2-u^2}}=\sin^{-1}\!\left(\frac{u}{a}\right)+C$$
$$\int \frac{du}{a^2-u^2}=\frac{1}{2a}\ln\left|\frac{a+u}{a-u}\right|+C$$
$$\int \frac{du}{\sqrt{u^2-a^2}}=\ln\left|u+\sqrt{u^2-a^2}\right|+C$$
Trigonometric Substitution
Case 1: Expressions of the Form $\sqrt{a^2 - x^2}$
$$x = a\sin\theta$$
$$dx = a\cos\theta\,d\theta$$
$$\sqrt{a^2 - x^2} = a\cos\theta$$
Case 2: Expressions of the Form $\sqrt{a^2 + x^2}$
$$x = a\tan\theta$$
$$dx = a\sec^2\theta\,d\theta$$
$$\sqrt{a^2 + x^2} = a\sec\theta$$
Case 3: Expressions of the Form $\sqrt{x^2 - a^2}$
$$x = a\sec\theta$$
$$dx = a\sec\theta\tan\theta\,d\theta$$
$$\sqrt{x^2 - a^2} = a\tan\theta$$
Back-Substitution Using Right Triangles
$$\sin\theta=\frac{x}{a}\qquad \cos\theta=\frac{\sqrt{a^2-x^2}}{a}$$
$$\tan\theta=\frac{x}{a}\qquad \sec\theta=\frac{\sqrt{a^2+x^2}}{a}$$
$$\sec\theta=\frac{x}{a}\qquad \tan\theta=\frac{\sqrt{x^2-a^2}}{a}$$
Hyperbolic Functions
$$\sinh x=\frac{e^x-e^{-x}}{2}$$
$$\cosh x=\frac{e^x+e^{-x}}{2}$$
$$\tanh x=\frac{\sinh x}{\cosh x}$$
$$\coth x=\frac{\cosh x}{\sinh x}$$
$$\operatorname{sech}x=\frac{1}{\cosh x}$$
$$\operatorname{csch}x=\frac{1}{\sinh x}$$
Key Hyperbolic Identities
$$\cosh^2 x-\sinh^2 x=1$$
$$1-\tanh^2 x=\operatorname{sech}^2 x$$
$$\coth^2 x-\operatorname{csch}^2 x=1$$
Derivatives of Hyperbolic Functions
$$\frac{d}{dx}\bigl(\sinh u\bigr)=\cosh u\,\frac{du}{dx}$$
$$\frac{d}{dx}\bigl(\cosh u\bigr)=\sinh u\,\frac{du}{dx}$$
$$\frac{d}{dx}\bigl(\tanh u\bigr)=\operatorname{sech}^2 u\,\frac{du}{dx}$$
$$\frac{d}{dx}\bigl(\coth u\bigr)=-\operatorname{csch}^2 u\,\frac{du}{dx}$$
$$\frac{d}{dx}\bigl(\operatorname{sech}u\bigr)=-\operatorname{sech}u\,\tanh u\,\frac{du}{dx}$$
$$\frac{d}{dx}\bigl(\operatorname{csch}u\bigr)=-\operatorname{csch}u\,\coth u\,\frac{du}{dx}$$
Improper Integrals
$$\int_a^\infty f(x)\,dx=\lim_{b\to\infty}\int_a^b f(x)\,dx$$
$$\int_{-\infty}^b f(x)\,dx=\lim_{a\to-\infty}\int_a^b f(x)\,dx$$
$$\int_a^b f(x)\,dx=\lim_{t\to c^-}\int_a^t f(x)\,dx+\lim_{t\to c^+}\int_t^b f(x)\,dx$$
Applications: Area Under a Curve
$$A=\int_a^b f(x)\,dx$$
Applications: Area Between Curves
$$A=\int_a^b \bigl(f(x)-g(x)\bigr)\,dx$$
$$A=\int_c^d \bigl(F(y)-G(y)\bigr)\,dy$$
Applications: Volume by Cross-Sections
$$V=\int_a^b A(x)\,dx$$
Applications: Volume of Revolution (Disk)
$$V=\pi\int_a^b \bigl(R(x)\bigr)^2\,dx$$
Applications: Volume of Revolution (Washer)
$$V=\pi\int_a^b \Big(\bigl(R(x)\bigr)^2-\bigl(r(x)\bigr)^2\Big)\,dx$$
Applications: Volume of Revolution (Shell)
$$V=2\pi\int_a^b \bigl(\text{radius}\bigr)\bigl(\text{height}\bigr)\,dx$$
$$V=2\pi\int_c^d \bigl(\text{radius}\bigr)\bigl(\text{height}\bigr)\,dy$$
Applications: Arc Length
$$L=\int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx$$
$$L=\int_c^d \sqrt{1+\left(\frac{dx}{dy}\right)^2}\,dy$$
Applications: Surface Area of Revolution
$$S=2\pi\int_a^b y\,\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx$$
$$S=2\pi\int_a^b x\,\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx$$
Pappus' Centroid Theorem I (Surface Area of Revolution)
$$S=2\pi d\,L$$
$$S=2\pi \bar{r}\,L$$
Pappus' Centroid Theorem II (Volume of Revolution)
$$V=2\pi d\,A$$
$$V=2\pi \bar{r}\,A$$
Centroid Distance to Axis
$$d=\bar{r}$$
Area Moments of Inertia (Second Moments of Area)
$$I_x=\int y^2\,dA$$
$$I_y=\int x^2\,dA$$
$$I_{xy}=\int xy\,dA$$
Polar Moment of Inertia
$$J_O=\int r^2\,dA$$
$$J_O=I_x+I_y$$
Parallel-Axis Theorem (Area)
$$I_x=I_{\bar{x}}+A d^2$$
$$I_y=I_{\bar{y}}+A d^2$$
Parallel-Axis Theorem (Polar)
$$J_O=J_C+A d^2$$
Radius of Gyration
$$k_x=\sqrt{\frac{I_x}{A}}$$
$$k_y=\sqrt{\frac{I_y}{A}}$$
$$k_O=\sqrt{\frac{J_O}{A}}$$
Mass Moment of Inertia
$$I=\int r^2\,dm$$
$$I_x=\int (y^2+z^2)\,dm$$
$$I_y=\int (x^2+z^2)\,dm$$
$$I_z=\int (x^2+y^2)\,dm$$
Parallel-Axis Theorem (Mass)
$$I_O=I_C+M d^2$$
Applications: Work
$$W=\int_a^b F(x)\,dx$$
$$W=\int_C \vec{F}\cdot d\vec{r}$$
$$W=\int_a^b F(x)\,\frac{dx}{dt}\,dt$$
Applications: Pumping Work
$$dW=\rho g\,A(y)\,d(y)\,dy$$
$$W=\rho g\int_{y_1}^{y_2} A(y)\,d(y)\,dy$$
Applications: Hydrostatic Pressure
$$p=\rho g h$$
Applications: Hydrostatic Force on a Vertical Plate
$$F=\int p\,dA$$
$$F=\rho g\int h\,dA$$
$$F=\rho g\int_{y_1}^{y_2} h(y)\,w(y)\,dy$$
Applications: Center of Mass (Lamina)
$$M=\iint_R \rho(x,y)\,dA$$
$$\bar{x}=\frac{1}{M}\iint_R x\,\rho(x,y)\,dA$$
$$\bar{y}=\frac{1}{M}\iint_R y\,\rho(x,y)\,dA$$
$$\text{If }\rho\text{ is constant: }M=\rho A,\;\bar{x}=\frac{1}{A}\iint_R x\,dA,\;\bar{y}=\frac{1}{A}\iint_R y\,dA$$
Applications: Centroid Under $y=f(x)$
$$A=\int_a^b f(x)\,dx$$
$$\bar{x}=\frac{1}{A}\int_a^b x\,f(x)\,dx$$
$$\bar{y}=\frac{1}{A}\cdot\frac{1}{2}\int_a^b \bigl(f(x)\bigr)^2\,dx$$
Applications: Centroid Between $y=f(x)$ and $y=g(x)$
$$A=\int_a^b \bigl(f(x)-g(x)\bigr)\,dx$$
$$\bar{x}=\frac{1}{A}\int_a^b x\bigl(f(x)-g(x)\bigr)\,dx$$
$$\bar{y}=\frac{1}{A}\cdot\frac{1}{2}\int_a^b \Big(\bigl(f(x)\bigr)^2-\bigl(g(x)\bigr)^2\Big)\,dx$$
Average Value of a Function
$$f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx$$
