**– 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.

\( a\) = \(\frac{dy}{dt} \) = \(\frac{d^2y}{dt^2} \)