Master Integration in Mathematics
Your comprehensive guide to understanding and solving integration problems
Basic Concepts
What is Integration?
Integration is the process of finding the area under a curve or the accumulation of quantities over time.
\[ \int f(x) dx = F(x) + C \]
Integration Techniques
Power Rule
For any real number n ≠ -1:
\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \]
Substitution
Let u = g(x), then:
\[ \int f(g(x))g'(x) dx = \int f(u) du \]
Integration by Parts
For functions u and v:
\[ \int u dv = uv - \int v du \]
Integration Formulas
Basic Integrals
\[ \int k dx = kx + C \]
\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) \]
\[ \int \frac{1}{x} dx = \ln|x| + C \]
\[ \int e^x dx = e^x + C \]
\[ \int a^x dx = \frac{a^x}{\ln a} + C \]
Trigonometric Functions
\[ \int \sin x dx = -\cos x + C \]
\[ \int \cos x dx = \sin x + C \]
\[ \int \tan x dx = -\ln|\cos x| + C \]
\[ \int \cot x dx = \ln|\sin x| + C \]
\[ \int \sec x dx = \ln|\sec x + \tan x| + C \]
\[ \int \csc x dx = -\ln|\csc x + \cot x| + C \]
Inverse Trigonometric Functions
\[ \int \frac{1}{\sqrt{1-x^2}} dx = \arcsin x + C \]
\[ \int \frac{1}{1+x^2} dx = \arctan x + C \]
\[ \int \frac{1}{|x|\sqrt{x^2-1}} dx = \arcsec x + C \]
Hyperbolic Functions
\[ \int \sinh x dx = \cosh x + C \]
\[ \int \cosh x dx = \sinh x + C \]
\[ \int \tanh x dx = \ln|\cosh x| + C \]
\[ \int \coth x dx = \ln|\sinh x| + C \]
Special Integrals
\[ \int \ln x dx = x\ln x - x + C \]
\[ \int \frac{1}{x^2 + a^2} dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C \]
\[ \int \frac{1}{\sqrt{a^2 - x^2}} dx = \arcsin\left(\frac{x}{a}\right) + C \]
\[ \int \frac{1}{x^2 - a^2} dx = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C \]
Integration by Parts
\[ \int u dv = uv - \int v du \]
Common applications:
\[ \int x^n e^x dx = x^n e^x - n\int x^{n-1} e^x dx \] \[ \int x^n \ln x dx = \frac{x^{n+1}}{n+1}\ln x - \frac{x^{n+1}}{(n+1)^2} + C \]Trigonometric Substitution
For \(\sqrt{a^2 - x^2}\):
\[ x = a\sin\theta \]For \(\sqrt{a^2 + x^2}\):
\[ x = a\tan\theta \]For \(\sqrt{x^2 - a^2}\):
\[ x = a\sec\theta \]Partial Fractions
For rational functions:
\[ \int \frac{P(x)}{Q(x)} dx \]where P(x) and Q(x) are polynomials
Practice Problems
Problem 1
Evaluate the following integral:
\[ \int x^2 dx \]