– Fundamental theorem of Calculus
If a function f(x) is continuous over an interval [a,b] and the function F(x) is defined by
F(x) = \(\int _a^b f(x)dx\)
Then F'(x) = f(x) over the interval [a , b]
– Integration of \(x^n\) (n ≠ -1) and some standard integrals
\(\int x^n dx\) = \(\frac{x^{n+1}}{n+1} + C\) (n ≠ -1)
\(\int x^2 dx\) = \(\frac{x^3}{3} + C\)
\(\int x^5 dx\) = \(\frac{x^6}{6} + C\)
\(\int e^{kx} dx\) = \(\frac{e^{kx}}{k} + C\)
\(\int {\frac{dx}{x}}\) = ln |(x)| + C
\(\int sin(ax)dx\) = –\(\frac{cos(ax)}{a} + C \)
\(\int cos(ax)dx\) = –\(\frac{sin(ax)}{a} + C \)
– Definite Integrals
A definite is used th calculate the area under the curve f(x) in the interval [a,b] as
A = \(\int _a^b f(x)dx\) = F(b) – F(a)
– Integration as limit as a sum
\(\int _a^b f(x)dx\) = \(\lim_{x\to\infty}\sum_{a^6}f(x)∆x\)
– Integration by substitution and Integration by parts
\(\int {\frac{f'(x)}{f(x)}}dx\) = ln f(x) + C
\(\int u' (x)cos(u(x))\) = \(sin(u(x)) + C \)
\(\int u' (x)sin(u(x))\) = \(-cos(u(x)) + C \)
\(\int{\frac{ u'(x)dx}{cos^2u(x)}}\) = \(\int u' (x)(1 + tan^2u(x))dx\) = \(tan(u(x)) + C\)
– Integration using partial fractions
\(\int{\frac{dx}{1-x^2}}\) = \(\int{\frac{dx}{2(1+x)}}\) + \(\int{\frac{dx}{2(1-x)}}\)
– Solving first order differential equations
\(\frac{dy}{dx}\) = \( 3x + 2\)
\( dy\) = \(( 3x + 2)dx\)
\(\int dy\) = \(\int (3x + 2)dx\)
\( y\) = \(\frac{3x^2}{2} + 2x + C\)
– Interpretation of the solution of a differential equation
Many problems in kinematics (acceleration, velocity, distance) can be solved using differential equations.
\( a\) = \(\frac{dy}{dt} \) = \(\frac{d^2y}{dt^2} \)