Integral

Integral
If the function F(x) is an antiderivative of f(x) then the expression F(x)+C where C is an arbitrary constant, is called the indefinite integral of f(x) and we write
??f(x) dx=F(x)+C
The integral of the power function
If u is a function of x, then
1.???u^n du=u^(n+1)/(n+1)?+c ;n?R , n?-1
2.??du/u=ln?|u|+c
Examples
1.???(3x+1) dx=( 1 )/3 ??3 (3x+1)^(( 1 )/2) dx=( 1 )/3 (3x+1)^(( 3 )/2)× ( 2 )/3+c=( 2 )/9 (3x+1)^(( 3 )/2)+c
2. ??xdx/?(x^2-3)=( 1 )/( 2 ) ???2x(x^2-3)^((-1)?2) dx?=( 1 )/( 2 ) (x^2-3)^(1?2)×2+c=?(x^2-3)+c
3. ??xdx/(x^2-3)=( 1 )/( 2 ) ln?|x^2-3|+c
The integrals of trigonometric functions
If u is a function of x, then
1.???sin?u du?=-cos?u+c 2.??cos?u du=sin?u+c
3.???sec^2?u du?=tan?u+c 4.??csc^2?u du=-cot?u+c
5.???sec?u tan?u du?=sec??u+c? 6.???csc?u cot?u du?=-csc?u+c

7.???tan?u du?=-ln?|cos?u |+c=ln?|sec?u |+c
8.???cot?u du?=ln?|sin?u |+c=-ln?|csc?u |+c


Examples
4.???(1+cos?2x ) dx= ???(2 cos^2?x ) dx= ???2 cos?x dx=?2 sin?x+c
5.??sin?2x/?(1+cos?2x ) dx=???(1+cos?2x )^((-1)?2) sin?2x ? dx=?(1+cos?2x )+c
6.??sin?2x/(1+cos?2x ) dx=( 1 )/( 2 ) ??(2 sin?2x)/(1+cos?2x ) dx=( 1 )/( 2 ) ln?(cos?2x )+c
The integrals of exponential functions
If u is a function of x, then
1.???a^u du?=a^u/ln?a +c ; a>0,a?1 2.??e^u du=e^u+c
Examples
7.??e^3x dx=( 1 )/( 3 ) e^3x+c
8.??(2e^(-2x)+3x^2 ) dx=-e^(-2x)+x^3+c
The integrals concerning the inverse trigonometric functions
1.??du/?(a^2-u^2 )=sin^(-1)??( u )/a?+c_1=-cos^(-1)??( u )/a?+c_(2 )
2.???du/(a^2+u^2 )=? ( 1 )/a tan^(-1)??( u )/a?+c_1=?( @-( 1 )/a cot^(-1)??( u )/a?+c_2 )
3.??du/(u ?(u^2-a^2 ))=( 1 )/a sec^(-1)??( u )/a?+c_1=-( 1 )/a csc^(-1)??( u )/a?+c_2
Examples
9.??dx/?(2-x^2 )=sin^(-1)??( x )/?2?+c
10.???dx/(x^2+2x+2)=? ??dx/(x^2+2x+1+1)
=???dx/((x+1)^2+1)=? tan^(-1)?(x+1)+c
11.??dx/(x ?(x^2-3))=( 1 )/?(3 ) sec^(-1)??( x )/?(3 )?+c
12.???(2x+7)/(x^2+4x+5) dx?=???(2x+4+3)/(x^2+4x+5) dx?
=???(2x+4)/(x^2+4x+5) dx?+???3/(x^2+4x+4+1) dx?
=ln?|x^2+4x+5|+???3/((x+2)^2+1) dx?
=ln?|x^2+4x+5|+3 tan^(-1)?(x+2)+c


Evaluate the following integrals
1. ??(x+?x)dx 2. ????(3x-2) dx?
3. ??xdx/?(1-?4x?^2 ) 4. ??xdx/(x^2+3)
5. ??dx/?(1-?4x?^2 ) 6. ??dx/(x^2+3)
7.???(1-cos?4x ) dx 8.???(x-1)^2/(x?x) dx?
9.??dx/(x^2-2x+5) 10.??(2x-5)dx/(x^2-4x+5)