Integration Cheatsheet

Cheatsheet Content

1. Basic Integration Rules Constant Rule: $\int k \, dx = kx + C$ Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, for $n \neq -1$ Logarithmic Rule: $\int \frac{1}{x} \, dx = \ln|x| + C$ Exponential Rule: $\int e^x \, dx = e^x + C$ General Exponential Rule: $\int a^x \, dx = \frac{a^x}{\ln a} + C$, for $a > 0, a \neq 1$ Constant Multiple Rule: $\int k f(x) \, dx = k \int f(x) \, dx$ Sum/Difference Rule: $\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$ 2. Trigonometric Integrals $\int \sin x \, dx = -\cos x + C$ $\int \cos x \, dx = \sin x + C$ $\int \sec^2 x \, dx = \tan x + C$ $\int \csc^2 x \, dx = -\cot x + C$ $\int \sec x \tan x \, dx = \sec x + C$ $\int \csc x \cot x \, dx = -\csc x + C$ $\int \tan x \, dx = \ln|\sec x| + C = -\ln|\cos x| + C$ $\int \cot x \, dx = \ln|\sin x| + C$ $\int \sec x \, dx = \ln|\sec x + \tan x| + C$ $\int \csc x \, dx = -\ln|\csc x + \cot x| + C$ 3. Inverse Trigonometric Integrals $\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C$ $\int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$ $\int \frac{1}{x\sqrt{x^2 - a^2}} \, dx = \frac{1}{a} \operatorname{arcsec}\left(\frac{|x|}{a}\right) + C$ 4. Integration Techniques 4.1. U-Substitution Identify $u = g(x)$, then $du = g'(x) \, dx$. Rewrite the integral in terms of $u$ and $du$. Integrate with respect to $u$. Substitute back $g(x)$ for $u$. Example: $\int 2x \cos(x^2) \, dx$. Let $u = x^2$, $du = 2x \, dx$. Integral becomes $\int \cos u \, du = \sin u + C = \sin(x^2) + C$. 4.2. Integration by Parts Formula: $\int u \, dv = uv - \int v \, du$ Choose $u$ and $dv$ using LIATE (Logarithmic, Inverse Trig, Algebraic, Trig, Exponential) for $u$. Differentiate $u$ to find $du$, integrate $dv$ to find $v$. Apply the formula. Example: $\int x e^x \, dx$. Let $u=x, dv=e^x \, dx$. Then $du=dx, v=e^x$. So, $xe^x - \int e^x \, dx = xe^x - e^x + C$. 4.3. Trigonometric Substitution For integrals involving: $\sqrt{a^2 - x^2}$: Let $x = a \sin \theta$, $dx = a \cos \theta \, d\theta$. $\sqrt{a^2 - x^2} = a \cos \theta$. $\sqrt{a^2 + x^2}$: Let $x = a \tan \theta$, $dx = a \sec^2 \theta \, d\theta$. $\sqrt{a^2 + x^2} = a \sec \theta$. $\sqrt{x^2 - a^2}$: Let $x = a \sec \theta$, $dx = a \sec \theta \tan \theta \, d\theta$. $\sqrt{x^2 - a^2} = a \tan \theta$. After integration, convert back to $x$ using a right triangle. 4.4. Partial Fraction Decomposition Used for integrating rational functions $\frac{P(x)}{Q(x)}$ where degree of $P(x)$ Steps: Factor the denominator $Q(x)$. Set up the partial fraction decomposition based on the factors: Linear factor $(ax+b)$: $\frac{A}{ax+b}$ Repeated linear factor $(ax+b)^n$: $\frac{A_1}{ax+b} + \frac{A_2}{(ax+b)^2} + \dots + \frac{A_n}{(ax+b)^n}$ Irreducible quadratic factor $(ax^2+bx+c)$: $\frac{Ax+B}{ax^2+bx+c}$ Repeated irreducible quadratic factor $(ax^2+bx+c)^n$: $\frac{A_1x+B_1}{ax^2+bx+c} + \dots + \frac{A_nx+B_n}{(ax^2+bx+c)^n}$ Solve for the unknown constants (A, B, ...). Integrate the resulting simpler fractions. 5. Definite Integrals Fundamental Theorem of Calculus (Part 2): $\int_a^b f(x) \, dx = F(b) - F(a)$, where $F'(x) = f(x)$. Properties: $\int_a^a f(x) \, dx = 0$ $\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx$ $\int_a^b k f(x) \, dx = k \int_a^b f(x) \, dx$ $\int_a^b [f(x) \pm g(x)] \, dx = \int_a^b f(x) \, dx \pm \int_a^b g(x) \, dx$ $\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$ 6. Improper Integrals Type 1 (Infinite Limits): $\int_a^\infty f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx$ $\int_{-\infty}^b f(x) \, dx = \lim_{a \to -\infty} \int_a^b f(x) \, dx$ $\int_{-\infty}^\infty f(x) \, dx = \int_{-\infty}^c f(x) \, dx + \int_c^\infty f(x) \, dx$ (must converge independently) Type 2 (Discontinuous Integrand): If $f(x)$ is discontinuous at $b$: $\int_a^b f(x) \, dx = \lim_{t \to b^-} \int_a^t f(x) \, dx$ If $f(x)$ is discontinuous at $a$: $\int_a^b f(x) \, dx = \lim_{t \to a^+} \int_t^b f(x) \, dx$ If $f(x)$ is discontinuous at $c \in (a,b)$: $\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$ An improper integral converges if the limit exists and is finite; otherwise, it diverges. 7. Applications of Integration Area between Curves: $A = \int_a^b |f(x) - g(x)| \, dx$ Volume of Revolution: Disk Method: $V = \pi \int_a^b [R(x)]^2 \, dx$ (revolving around x-axis) Washer Method: $V = \pi \int_a^b ([R(x)]^2 - [r(x)]^2) \, dx$ (revolving around x-axis) Shell Method: $V = 2\pi \int_a^b x f(x) \, dx$ (revolving around y-axis) Arc Length: $L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx$ Surface Area of Revolution: $S = 2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2} \, dx$ (revolving $y=f(x)$ around x-axis) Work: $W = \int_a^b F(x) \, dx$ (force function) Average Value of a Function: $f_{avg} = \frac{1}{b-a} \int_a^b f(x) \, dx$ 8. Common Integration Errors Forgetting the $+C$ for indefinite integrals. Distributing integrals over products/quotients: $\int f(x)g(x) \, dx \neq (\int f(x) \, dx)(\int g(x) \, dx)$. Incorrectly applying power rule for $n=-1$. Errors in algebraic manipulation during substitution or partial fractions. Not changing limits of integration for definite integrals when using u-substitution.