Basic Integration Rules
- \(\int a \, dx = ax + C\)
- \(\int x^a \, dx = \frac{x^{a+1}}{a + 1} + C\) (for \(a \neq -1\))
- \(\int (f(x))^a f'(x)dx = \frac{(f(x))^{a+1}}{a + 1} + C\) (for \(a \neq -1\))
- \(\int \frac{dx}{x} = \ln |x| + C\) (for \(x \neq 0\))
- \(\int \frac{f'(x)}{f(x)} dx = \ln |f(x)| + C\)
Exponential Integrals
- \(\int e^{ax + b} dx = \frac{1}{a} e^{ax + b} + C\) (for \(a \neq 0\))
- \(\int a^{\alpha x + \beta} dx = \frac{a^{\alpha x + \beta}}{\alpha \ln(a)} + C\) (for \(a > 0, a \neq 1, \alpha \neq 0\))
Trigonometric Integrals
- \(\int \sin(ax + b)dx = -\frac{1}{a} \cos(ax + b) + C\)
- \(\int \cos(ax + b)dx = \frac{1}{a} \sin(ax + b) + C\)
- \(\int \sec^2(ax + b)dx = \frac{1}{a} \tan(ax + b) + C\)
- \(\int \csc^2(ax + b)dx = -\frac{1}{a} \cot(ax + b) + C\)
- \(\int \sec x \tan x \, dx = \sec x + C\)
- \(\int \csc x \cot x \, dx = -\csc x + C\)
- \(\int \tan x \, dx = -\ln|\cos x| + C = \ln|\sec x| + C\)
- \(\int \cot x \, dx = \ln|\sin x| + C\)
- \(\int \sec x \, dx = \ln|\sec x + \tan x| + C\)
- \(\int \csc x \, dx = -\ln|\csc x + \cot x| + C\)
Hyperbolic Integrals
- \(\int \sinh(ax + b)dx = \frac{1}{a} \cosh(ax + b) + C\)
- \(\int \cosh(ax + b)dx = \frac{1}{a} \sinh(ax + b) + C\)
- \(\int \operatorname{sech}^2(ax + b)dx = \frac{1}{a} \tanh(ax + b) + C\)
- \(\int \operatorname{csch}^2(ax + b)dx = -\frac{1}{a} \coth(ax + b) + C\)
- \(\int \tanh x \, dx = \ln(\cosh x) + C\)
- \(\int \coth x \, dx = \ln|\sinh x| + C\)
Inverse Trigonometric Integrals
- \(\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C\) (for \(a \neq 0\))
- \(\int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\left(\frac{x}{a}\right) + C\) (for \(a > 0\))
- \(\int \frac{dx}{\sqrt{x^2 + a^2}} = \ln \left| x + \sqrt{x^2 + a^2} \right| + C = \operatorname{arsinh}\left(\frac{x}{a}\right) + C\)
- \(\int \frac{dx}{\sqrt{x^2 - a^2}} = \ln \left| x + \sqrt{x^2 - a^2} \right| + C = \operatorname{arcosh}\left(\frac{x}{a}\right) + C\) (for \(x > a > 0\))
- \(\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln \left| \frac{a + x}{a - x} \right| + C = \frac{1}{a} \operatorname{artanh}\left(\frac{x}{a}\right) + C\) (for \(|x| < a\))
- \(\int \frac{dx}{x\sqrt{x^2 - a^2}} = \frac{1}{a} \operatorname{arcsec}\left(\frac{|x|}{a}\right) + C\) (for \(|x| > a > 0\))
Integration Techniques
Integration by Parts
- \(\int u \, dv = uv - \int v \, du\)
Substitution Rule for Definite Integrals
- \(\int_a^b f(g(x)) g'(x) \, dx = \int_{g(a)}^{g(b)} f(u) \, du\)
Fundamental Theorem of Calculus
- Part 1: \(\frac{d}{dx} \int_a^x f(t) \, dt = f(x)\)
- Part 2: \(\int_a^b f(x) \, dx = F(b) - F(a)\) where \(F'(x) = f(x)\)
- Leibniz Rule: \(\frac{d}{dx} \int_{u(x)}^{v(x)} f(t) \, dt = f(v(x)) \cdot v'(x) - f(u(x)) \cdot u'(x)\)
Properties of Definite Integrals
- Linearity: \(\int_a^b [cf(x) + g(x)] \, dx = c\int_a^b f(x) \, dx + \int_a^b g(x) \, dx\)
- Additivity: \(\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx\)
- Reversal: \(\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx\)
- Zero-width: \(\int_a^a f(x) \, dx = 0\)
- Even Function: \(\int_{-a}^a f(x) \, dx = 2\int_0^a f(x) \, dx\) (if \(f(-x) = f(x)\))
- Odd Function: \(\int_{-a}^a f(x) \, dx = 0\) (if \(f(-x) = -f(x)\))
- Average Value: \(f_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x) \, dx\)
Applications of Definite Integrals
- Area Between Curves: \(A = \int_a^b [f(x) - g(x)] \, dx\) (where \(f(x) \ge g(x)\))
- Volume by Disk Method: \(V = \pi \int_a^b [f(x)]^2 \, dx\)
- Volume by Washer Method: \(V = \pi \int_a^b \left([f(x)]^2 - [g(x)]^2\right) dx\)
- Volume by Shell Method: \(V = 2\pi \int_a^b x f(x) \, dx\) (revolving about \(y\)-axis)
- Arc Length: \(L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx\)
- Parametric Arc Length: \(L = \int_\alpha^\beta \sqrt{[x'(t)]^2 + [y'(t)]^2} \, dt\)
- Surface Area of Revolution: \(S = 2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2} \, dx\) (about \(x\)-axis)
Improper Integrals
- Type 1 (Infinite Interval): \(\int_a^\infty f(x) \, dx = \lim_{t \to \infty} \int_a^t f(x) \, dx\)
- Type 2 (Discontinuity at \(b\)): \(\int_a^b f(x) \, dx = \lim_{t \to b^-} \int_a^t f(x) \, dx\)
- \(p\)-Integral Test: \(\int_1^\infty \frac{dx}{x^p}\) converges if \(p > 1\), diverges if \(p \le 1\)
Trigonometric Identities
- Power Reduction: \(\sin^2 x = \frac{1 - \cos 2x}{2}\), \(\cos^2 x = \frac{1 + \cos 2x}{2}\)
- Product-to-Sum:
- \(\sin(mx)\cos(nx) = \frac{1}{2}[\sin((m+n)x) + \sin((m-n)x)]\)
- \(\sin(mx)\sin(nx) = \frac{1}{2}[\cos((m-n)x) - \cos((m+n)x)]\)
- \(\cos(mx)\cos(nx) = \frac{1}{2}[\cos((m-n)x) + \cos((m+n)x)]\)
- Pythagorean: \(\tan^2 x = \sec^2 x - 1\), \(1 + \tan^2 x = \sec^2 x\), \(1 + \cot^2 x = \csc^2 x\)
- Universal Substitution: \(t = \tan\frac{x}{2}\) gives:
- \(\sin x = \frac{2t}{1 + t^2}\)
- \(\cos x = \frac{1 - t^2}{1 + t^2}\)
- \(dx = \frac{2}{1 + t^2} dt\)
Hyperbolic Identities
- Power Reduction: \(\sinh^2 x = \frac{\cosh 2x - 1}{2}\), \(\cosh^2 x = \frac{\cosh 2x + 1}{2}\)
- Pythagorean: \(\cosh^2 x - \sinh^2 x = 1\), \(\tanh^2 x = 1 - \operatorname{sech}^2 x\)
- Definitions: \(\sinh x = \frac{e^x - e^{-x}}{2}\), \(\cosh x = \frac{e^x + e^{-x}}{2}\), \(\tanh x = \frac{\sinh x}{\cosh x}\)
Functions of Several Variables
Partial Derivatives
- Partial derivative w.r.t. \(x\): \(f_x(x_0,y_0) = \lim_{h\to0}\frac{f(x_0+h,y_0)-f(x_0,y_0)}{h}\)
- Chain Rule (1 parameter): \(\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}\)
- Chain Rule (2 parameters): \(\frac{\partial z}{\partial s} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial s}\)
- Implicit Differentiation (\(F(x,y)=0\)): \(\frac{dy}{dx} = -\frac{F_x}{F_y}\)
- Implicit Differentiation (\(F(x,y,z)=0\)): \(\frac{\partial z}{\partial x} = -\frac{F_x}{F_z}\), \(\frac{\partial z}{\partial y} = -\frac{F_y}{F_z}\)
- Linearization: \(\mathcal{L}(x,y) = f(x_0,y_0) + f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0)\)
- Laplace Equation (Harmonic): \(\frac{\partial^2 f}{\partial x_1^2} + \cdots + \frac{\partial^2 f}{\partial x_n^2} = 0\)
Directional Derivatives and Gradients
- Gradient: \(\nabla f = \left(\frac{\partial f}{\partial x_1}, \dots, \frac{\partial f}{\partial x_n}\right)^T\)
- Directional Derivative: \(D_\mathbf{u}f = \nabla f \cdot \mathbf{u}\) (where \(\|\mathbf{u}\| = 1\))
- Maximum rate of increase: \(\|\nabla f\|\) in direction \(\nabla f / \|\nabla f\|\)
Extreme Values
- Hessian / Discriminant: \(D = f_{xx}f_{yy} - (f_{xy})^2\)
- Second Derivative Test: local max if \(D>0,\,f_{xx}<0\); local min if \(D>0,\,f_{xx}>0\); saddle if \(D<0\)
Lagrange Multipliers
- One constraint: \(\nabla f = \lambda\nabla g\) and \(g(\mathbf{x})=0\)
- Lagrangian: \(\mathcal{L}(\mathbf{x},\lambda) = f(\mathbf{x}) - \lambda g(\mathbf{x})\)
- Multiple constraints: \(\nabla f = \sum_{i=1}^m \lambda_i \nabla g_i\) and \(g_i(\mathbf{x})=0\)
Gradient Descent
- Update rule: \(x^{k+1} = x^k - \alpha\nabla f(x^k)\)
Coordinate Systems
Polar and Cylindrical Coordinates
- Polar ↔︎ Cartesian (2D): \(x = r\cos\theta\), \(y = r\sin\theta\); \(r = \sqrt{x^2+y^2}\), \(\tan\theta = y/x\)
- Cylindrical ↔︎ Cartesian (3D): \(x = r\cos\theta\), \(y = r\sin\theta\), \(z = z\); \(r = \sqrt{x^2+y^2}\)
Spherical Coordinates
- Spherical ↔︎ Cartesian: \(x = \rho\sin\phi\cos\theta\), \(y = \rho\sin\phi\sin\theta\), \(z = \rho\cos\phi\); \(\rho = \sqrt{x^2+y^2+z^2}\)
- Spherical ↔︎ Cylindrical: \(r = \rho\sin\phi\), \(z = \rho\cos\phi\)
3D Geometry
- Distance formula: \(|P_1P_2| = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)
- Sphere: \((x-x_0)^2+(y-y_0)^2+(z-z_0)^2 = R^2\)
- Line (parametric): \(x=x_0+v_1t\), \(y=y_0+v_2t\), \(z=z_0+v_3t\)
- Plane: \(A(x-x_0)+B(y-y_0)+C(z-z_0)=0\), normal \(\mathbf{n}=(A,B,C)\)
Double Integrals
Definition and Fubini’s Theorem
- Double integral (definition): \(\iint_R f\,dA = \lim_{\|P\|\to0}\sum_{k=1}^n f(x_k,y_k)\Delta A_k\)
- Fubini (rectangle): \(\iint_R f\,dA = \int_c^d\int_a^b f\,dx\,dy = \int_a^b\int_c^d f\,dy\,dx\)
- Fubini (vertically simple): \(\iint_R f\,dA = \int_a^b\int_{g_1(x)}^{g_2(x)} f\,dy\,dx\)
- Fubini (horizontally simple): \(\iint_R f\,dA = \int_c^d\int_{h_1(y)}^{h_2(y)} f\,dx\,dy\)
- Area: \(\text{Area}(R) = \iint_R dA\)
- Average value: \(\text{aver}_R(f) = \dfrac{1}{\text{Area}(R)}\iint_R f\,dA\)
Change of Variables
- Jacobian (2D): \(J(u,v) = \dfrac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix}\partial x/\partial u & \partial x/\partial v\\ \partial y/\partial u & \partial y/\partial v\end{vmatrix}\)
- Change of variables: \(\iint_R f\,dx\,dy = \iint_G f(g,h)\,|J|\,du\,dv\)
- Polar coordinates (Jacobian \(J=r\)): \(\iint_R f\,dx\,dy = \iint_G f(r\cos\theta,r\sin\theta)\,r\,dr\,d\theta\)
- Area in polar: \(A = \dfrac{1}{2}\displaystyle\int_{\theta_1}^{\theta_2}[r(\theta)]^2\,d\theta\)
- Gaussian integral: \(\displaystyle\int_{-\infty}^{\infty}e^{-x^2}\,dx = \sqrt{\pi}\)
Triple Integrals
Definition and Fubini’s Theorem
- Triple integral: \(\iiint_D F\,dV = \lim_{\|P\|\to0}\sum_{k=1}^n F(x_k,y_k,z_k)\Delta V_k\)
- Fubini (general): \(\iiint_D F\,dV = \iint_R\!\left(\int_{f_1(x,y)}^{f_2(x,y)}F\,dz\right)dA\)
- Volume: \(\text{vol}(D) = \iiint_D dV\)
- Average value: \(\text{aver}_D(F) = \dfrac{1}{\text{vol}(D)}\iiint_D F\,dV\)
Change of Variables
- Jacobian (3D): \(J(u,v,w) = \dfrac{\partial(x,y,z)}{\partial(u,v,w)}\) (3×3 determinant)
- Change of variables: \(\iiint_D F\,dx\,dy\,dz = \iiint_G H\,|J|\,du\,dv\,dw\)
- Cylindrical coordinates (Jacobian \(J=r\)): \(\iiint_D F\,dV = \iiint_G H(r,\theta,z)\,r\,dr\,d\theta\,dz\)
- Spherical coordinates (Jacobian \(J=\rho^2\sin\phi\)): \(\iiint_D F\,dV = \iiint_G H(\rho,\phi,\theta)\,\rho^2\sin\phi\,d\rho\,d\phi\,d\theta\)
Lagrange Multipliers
- Lagrange condition (2D): \(\nabla f = \lambda\,\nabla g\), i.e., \(f_x = \lambda g_x\), \(f_y = \lambda g_y\), \(g(x,y) = 0\)
- Lagrange function: \(L(x,y,\lambda) = f(x,y) - \lambda\,g(x,y)\)
- Lagrange condition (3D): \(f_x = \lambda g_x\), \(f_y = \lambda g_y\), \(f_z = \lambda g_z\), \(g(x,y,z) = 0\)