Integration Formulas
May 3, 2025
Basic Formulas
- \displaystyle \intop kf(x) \space dx = k \intop \space f(x) \space dx, \space k \text{ is a constant} .
- \displaystyle \intop f(x) \space \pm \space g(x) \space d(x) = \intop f(x) \space dx \space \pm \space \intop g(x) \space dx
- \displaystyle\intop^b_a {f(x) \space dx = F(x)\vert^b_a} \space = \space F(b) - F(a) \text{ where } F(x) = \intop{f(x) \space dx}
- \displaystyle\intop^b_a kf(x) \space dx =k \intop ^b_a \space f(x) \space dx, \space k \text{ is a constant}.
- \displaystyle\intop ^b_a f(x) \space \pm \space g(x) \space dx = \intop^b_a f(x) \space dx \space \pm \intop ^b_a g(x) \space dx
- \displaystyle \intop ^a_a f(x) \space dx = 0
- \displaystyle \intop ^b_a f(x) \space dx = - \intop ^a_b f(x) \space dx
- \displaystyle\intop ^b_a f(x) \space dx = \intop ^c_a f(x) \space dx + \intop^b_c f(x) \space dx
Common Integrals
Polynomials
- \displaystyle\intop dx = x +C
- \displaystyle\intop x^n \space dx = \dfrac{x^{n+1}}{n +1} + C, \space n \neq -1
- \displaystyle\intop k \space dx = kx +C, \space k \text{ is a constant}.
- \displaystyle\intop x^{-1} \space dx = \intop \dfrac{1}{x}dx = \text {ln}|x| + C
Trigonometric Functions
- \displaystyle\intop \cos(u) \space du = \sin(u) +C
- \displaystyle\intop \sin(u) \space du =-\cos(u) +C
- \displaystyle\intop \sec^2(u) \space du = \tan(u) +C
- \displaystyle\intop \tan(u) \space du = -\text{ln}|\cos(u)| + C \\
- \displaystyle\intop \cot(u) \space du = -\text{ln}|csc(u)|+C \\
- \displaystyle\intop \sec(u) \space du = - \text{ln}|\sec(u) + \tan(u)| +C \\
- \displaystyle\intop \sin^2(u) \space du = \dfrac{u}{2} - \dfrac{\sin(2u)}{4} +C
- \displaystyle\intop \csc(u) \cot (u) \space du = - \csc(u) +C \\
- \displaystyle\intop \sec(u) \tan(u) \space du = \sec(u) +C \\
- \displaystyle\intop \csc^2(u) \space du = -\cot(u) + C \\
- \displaystyle\intop \tan(u) \space du = \text{ln}|\sec(u)| + C \\
- \displaystyle\intop \cot(u) \space du =\text{ln}|\sin(u)| + C \\
- \displaystyle\intop \csc(u) \space du == \text{ln}|\csc9u) -\cot(u)| + C \\
- \displaystyle\intop\cos^2(u) \space du = \dfrac{u}{2} + \dfrac{\sin(2u)}{4} + C
Exponential and Logarithmic Functions
- \displaystyle\intop e^u \space du = e^u + C \\
- \displaystyle\intop a^u \space du = \dfrac{a^u}{\text{ln}(a)} +C \\
- \displaystyle\intop \cos^2(u) \space du = \dfrac{u}{2} + \dfrac{\sin(2u)}{4} +C
Inverse Trigonometric Functions
- \displaystyle\intop \dfrac{1}{\sqrt {a^2 - u^2}}du = \sin^{-1}\Big({\dfrac{u}{a}}\Big) + C \\
- \displaystyle\intop \dfrac{1}{u\sqrt{u^2 -a^2}}du = \dfrac{1}{a}\sec^{-1} \Big({\dfrac{u}{a}}\Big) + C\\
- \displaystyle\intop \dfrac{1}{a^2+u^2}du = \dfrac{1}{a}\tan^{-1} \Big(\dfrac{u}{a} \Big) + C