Calculus Two Integral

Power reducing equation for sin^2x

(1/2)(1-cos^2x)

Differentiation equations to integrals for arctan x

(S) (1/(1+x^2))dx=arctan x + K

Integrals of inverse trigonometric functions equation for arcsec x

5.5 (S) arcsec x dx=x arcsec x - ln ( x + square root of (x^2- 1) + c

Integrals of inverse trigonometric functions equation for arccsc x

5.6 (S) arccsc x dx= x arccsc x + ln (x+square root of (x^2 - 1) +c

Integrals involving exponential sin equation

6.1 (S) e^ax sin bx dx= (e^ax / (a^2 +b^2)(a sin bx - b cos bx) +c

Integrals involving exponential cos equation

6.2 (S) e^ax sin bx dx= (e^ax / (a^2 - b^2)(b sin bx + b cos bx) +c

Integrals involving hyperbolic functions sinh x

7.1 (S) sinh x dx= cos x+c

Integrals involving hyperbolic functions cosh x

7.2 (S) cos x dx=sinh x+c

Integrals involving hyperbolic functions sech x and tanh x

7.3 (S) sech x tanh x dx=-sech x+c

Integrals involving hyperbolic functions csch x coth x

7.4 (S) csch x coth x dx=-Csch x+c

Integrals involving hyperbolic functions sech^2 x

7.5 (S) sech^2 x dx=tanh x+c

Integrals involving hyperbolic functions csch^2 x

7.6 (S) csch ^2 x dx=-coth x+x

How to find area under curve for a curve involving x axis

A=(S) x1^x2 (f(x)-h(x))dx

How to find area under curve for a curve involving y axis

A=(S) y1^y2(z(y)- x(y))dx

How to solve Integration by Parts (S) 3x e^x dx

u=x du/dx=1 v=e^x 3 (S) x e^x dx 3( x e^x - (S) 1e^x dx) 3x e^x - 3 e^x + K

How to solve integrals involving quadratics using competing the square

(S) (1/square root (x^2+12x+40)) dx First compete the square root of x^2+12x+40 and you get (x+6)^2+4 given the integral looks like this (S) (1/square root of (x+6)^2+4)) dx Factor out a 4 from under the square root (S) (1/(2 square root of (x/2 +3)^2 +1)) dx z=x/2 +3 hence 2 dz=dx (S) 1/square root of (z^2+1)) dx f'(x)=arcsinh(z) + K f'(x)=arcsinhx(x/2 +3) +K

Differentiation equations to integrals for for sinhx

(S) (1/square root of (1+x^2)) dx=arcsinh x + K

Differentiation equations to integrals for arcsin x

(S) (1/square root of (1-x^2)) dx=arcsin x + K

Differentiation equations to integrals for for cosh x

(S) (1/square root of (x^2-1))dx=arccosh x + K

Integral of Sum of functions equation

(S) (f(x) + g(x)) dx= (S)f(x) dx + (S) g(x) dx

Integral of Difference of Functions

(S) (f(x) - g(x)) dx= (S)f(x) dx - (S) g(x) dx

How to solve integrals involving sine or cosine and exponential

(S) sin(x) e^x dx use integral of parts let u= sin(x) dv/dx=e^x (S)sinx e^x dx=sinx e^x - (S) - cos(x) e^x dx apply integration (S)sinx e^x dx=sinx e^x -(cosx e^x -(S) - sinx e^x dx) simplify (S)sinx e^x dx=sinx e^x -cosx e^x -(S) sinx e^x dx Note that the term on the right is the integral we are trying to evalute hence the above may be 2(S) sinx e^x dx=sinx e^x -cosx e^x and the answer is (S) sinx e^x dx=(1/2)e^x (sin(x) - cos(x)) + C

How to solve integrals involving sinx with odd power (S) sin^5 x dx

(S) sin^5 x dx=(S)sin^4 sin x dx f(x)=(S)(1-cos^2 x)^2 sin x dx sin^2=1-cos^2x u=cosx du/dx=-sinx or du=-sinxdx (S) sin^5 x dx=-(S)(1-u^2)^2 f(x)=-(S)(u^4 -2u^2+1)du f(x)=-1/5 cos^5 x + 2/3cos^3 x -cos x +C

Table of Leplace transforms equation for e^-at cos bt

(s + a)/(s + a)^2 +b

Table of Leplace transforms equation for t cos at

(s^2 - a^2)/(s^2 + a^2)

Integral of elementary function equation for (S) dx

1.1 (S) dx= x+c

Integral of elementary function equation for (S) k dx

1.2 (S) k dx=k x + c where k is a constant

Integral of elementary function equation for power function

1.3 (S) x^n dx=x^n+1/(n+1) + c

Integral of elementary function equation for logs and natural logs

1.4 (S) (1/x) dx =ln absolute value of (x) +c

Table of Fourier Transforms Pairs f(w)

1/(a+jw) , (pie/j)(d(w- a) - d(w+ a)), f(w)= (S) top curve infinite bottom curve negative infinite e^ -jwt f(t) dt

Table of Leplace transforms equation for te^-at

1/(s+a)^2

Power reducing equation for sin^5x

1/16(sin5x-5 sin3x + 10sinx)

Power reducing equation for sin^6x

1/32(10-15cos2x+6cos4x - cos6x)

Power reducing equation for sin^3x

1/4(3 sinx -sin3x)

Power reducing equation for sin^4x

1/8(3-4cos2x + cos4x)

Table of Leplace transforms equation for 1

1=1/s

Table of Leplace transforms equation for t sin at

2 a s/(s^2 +a^2)^2

Integral of Elementary trigonometric functions for sin

2.1 (S) sin x dx= -cos x+c

Integral of Elementary trigonometric functions for cos

2.2 (S) cos x dx= sinx+c

Integral of Elementary trigonometric functions for tan

2.3 (S) tan x dx= ln absolute value of (sec x) + c

Integral of Elementary trigonometric functions for cot

2.4 (S) cot x dx= ln absolute value of (sin x) + c

Integral of Elementary trigonometric functions for sec

2.5 (S) sec x dx= ln absolute value of (sec x + tan x) +c

Integral of Elementary trigonometric functions for Csc

2.6 (S) Csc x dx= ln absolute value of (Csc x - Cot c) + c

Integral involving more than one trigonometric function for sec and tan

3.1 (S) sec x tan x dx= sec x+ c

Integral of Elementary trigonometric functions for Csc and cot

3.2 (S) Csc x Cot x dx= -Csc x + c

Integral of Elementary trigonometric functions for sin and sin

3.3 (S) sin mx sin nx dx= -sin((m +n)x)/2(m+ n) + sin ((m +n)x)/2(m-n) +c with m not equal to n

Integral of Elementary trigonometric functions for cos cos

3.4 (S) cos mx cos nx dx= sin ((m + n)x)/2(m + n) +sin ((m-n)x)/2(m + n) + c with m not equal to n

Integral of Elementary trigonometric functions for sin and cos

3.5 (S) sin mx cos nx dx= - cos((m+n)x)/2(m+n) - cos((m -n)x)/2(m-n) + c with m not equal to n

Integrals involving exponential and logarithmic functions equation for e^x

4.1 (S) e^x dx=e^x +c

Integrals involving exponential and logarithmic functions equation for a^x a is for a constant

4.2 (S) a^x dx=a^x dx=a^x/ln a+ c

Integrals involving exponential and logarithmic functions equation for ln x

4.3 (S) ln x dx= x ln x - x+ c

Integrals of inverse trigonometric functions equation for arcsin x

5.1 (S)arcsin x dx=x arcsin x + square root of (1-x^2) + c

Integrals of inverse trigonometric functions equation for arccos x

5.2 (S) arccos x dx= x arccos x - square root of (1-x^2) + c

Integrals of inverse trigonometric functions equation for arctan x

5.3 (S) arctan x dx= x arctan x - ln ( square root of (1+x^2) +c

Integrals of inverse trigonometric functions equation for arccot x

5.4 (S) arccot x dx= x arcot x + ln square root of (1+x^2) +c

How to solve integration using partial fractions (S) (-5x+11)/(x^2+x-2) dx

First solve using partial function decomposition (-5x+11)/(x^2+x-2)= 2/(x-1) - 7/(x+2) and integral (S) 2/(x-1) dx - (S) 7/(x+2) dx=2ln absolute val (x-1) - 7ln abs (x+2) + C

Properties of integrals for k f(x)

S k f(x) dx= k (S) f(x) dx

How to solve partial function decompositions (2x+5)/(x^2 - x - 2)

Start by factoring the denominator (x^2 - x-2)= (x -2) (x+1) and (2x+5)/(x^2 - x - 2)= A/ (x -2) + B/(x+1) +... C/(denominator^4,5,6....) multiply each side by denominator 2x+5=A(x+1) + B(x-2) Expand the right side and group like terms 2x+5=x(A+B) + A-2B and 2=A+B 5=A-2B Solve with elimination or substitution A=3 and B=-1 (2x+5)/(x^2 - x - 2)= 3/ (x -2) - 1/(x+1)

How to evaluate integrals (S) x^3 e^x^4 dx

Use integral of substitution let u=x^4 du/dx=4x^3 which leads to 1/4du=x^3dx and the integral can be written as (S) 1/4 e^u du f(x)=1/4 e^u + c and then substitute u for u=x^4 and you get f(x)=(1/4) e^x^4 + c

How to solve the volume of a solid revolution when only given one function of x f(x)=r^2 - x^2 x1=-r x2=r

V=(S) -r^r pie(r^2 -x^2) dx V=p(r^2 x-x^3)) -r^r V=p(r^3-r^3/3 -(-r^3 + r^3 /3) V=4/3 pie r^3

How to solve the volume of a solid revolution when given two functions related to y y1=0 y2=1 z(y)=-y+2 w(y)=y

V=(S) 0^1 pie((-y+2)^2 - y^2) dy V=(S) 0^1 p(-4y+4)dy V=p(-2y^2 +4) 0^1 V=2pie

Equation for the volume of a solid revolution when you only given one function of x

V=(S) x1^x2 pie(f(x))^2 dx

Equation for the volume of a solid revolution when you given two functions relating to x

V=(S) x1^x2pie(f(x)^2 - h(x)^2) dx

Equation for the volume of a solid revolution when you only given one function of y

V=(S) y1^y2 pie(z(y))^2 dy

Equation for the volume of a solid revolution when you are given two functions relating to y

V=(S) y1^y2 pie(z(y)^2 - w (y)^2) dy

Table of Leplace transforms equation for sin a t

a/(s^2 + a^2)

Table of Leplace transforms equation for sinh a t

a/(s^2 - a^2)

Table of Leplace transforms equation for 1-cos at

a^2/s(s^2 + a^2)

Table of Leplace transforms equation for(1/t)(sin a t)

arctan(a/s)

Table of Leplace transforms equation for e^-at sin b t

b/(s+a)^2 + b^2

Table of Leplace transforms equation for e^-at

e^-at=1/(s+a)

Table of Leplace transforms equation for f(s)

f(s)=(E)(F(t)) and f(s)=L(f(t)= (S) v at the top 0 at the bottom e^-stf(t) dt

Table of Leplace transforms equation for f(t)

f(t)= (E)^-1 ((curved E)) (F(s))

How to solve integral of Difference of functions h(x)=(S) (2- 1/a) dx

f(x)= 2 g(x)=-1/x h'(x)=(S) 2 dx - (S) (1/x) dx h'(x)=(S) 2x - ln absolute value of (x) +c

How to solve the area under a curve (Incomplete)

f(x)=(-x^2+6) h(x)=(x^2-2x+2) set the two equations equal to each other and factor and you get x=(-1,2) A=(S)=-1^2((-x^2+6) -(x^2-2x+2)) dx A=(S) (-2x^2+2x+4)dx A=(-2/3x^3+^2+4x) -1^2 A=9 unsure how

How to solve integrals involving sinx with even power (S)(sin^2x -16 sin^6 x)dx

f(x)=(S) ((1/2)(1-cos2x)-16(1/32)(10-15cos2x+6 cos(4x)-cos(6x))dx f(x)=1/2(S) (1-cos2x-10+15cos2x -6 cos4x +cos6x )dx f(x)=1/2(-9x-1/2sin2x+(1/2) 15sin2x - (1/4)sin4x + 1/6 sin 6x) + C

Integration by Substitution equation

f(x)=(S) ((the top of the S like integral is b the last number and the bottom is the first)) f(g(x)) g'(x)) dx= (S) (((g(b) last g(a) first)) f(u) du and K is a constant of the integration

How to solve integrals involving sinx with even power (S)sin^2x dx

f(x)=(S)1/2(1-cos^2x)dx f(x)=1/2(S)dx-1/2(S)cos2x dx f(x)=1/2x (1/2)(1/2)sin2x f(x)=x/2-1/4sin2x+C

Table of Leplace transforms equation for (2/t)(t-cos a t)

ln((s^2 + a^2)/s^2)

Table of Leplace transforms equation for (2/t)(t- cosh a t)

ln((s^2 - a^2)/s^2)

Table of Leplace transforms equation for cos a t

s/(s^2 + a^2)

Table of Leplace transforms equation for cosh a t

s/(s^2 - a^2)

How to solve integrals involving sinx and cosx with odd power like sin^3(x)cos^2(x)dx

sin^3(x)cos^2(x)dx=sin^2x cos^2x sinx dx sin^2=cos^2-1 f(x)=(S) (1-cos^2 x cos^2 x sin x dx let u=cosx du/dx=-sinx or -du=sinxdx sin^3(x)cos^2(x)dx=-(S)(1-u^2) u^2 du f(x)=(S) u^4-u^2 du f(x)=(1/5)u^5 - 1/3u^3 + C f(x)=1/5cos^5x -1/3cos^3x +C

How to solve the area under a curve for a triangle

solve for the triangle area A=1/2bh A=4 b=2 h=4 f(x)=2x h(x)=0 A=(S) 0^2 (2x-0)dx=2 A=(S) 0^2 x dx=(2x^2/2)0^2 A=4

Table of Leplace transforms equation for t

t=1/s^2

Table of Leplace transforms equation for t^-1/2

t^-1/2=(pie/s)^1/2

Table of Leplace transforms equation for t^1/2 or square root of t

t^1/2=pie^1/2/2a^3/2

Table of Leplace transforms equation for t^n

t^n=n!/s^n+1 (n=1,2,3....)

Integration by Parts equation

u (du/dx) dx= uv - (S) (du/dx) v dx K is a constant of the integration

Table of Fourier Transforms Pairs f(t)

u(t) e^-at , a greater than zero 1 for -a less than t less than a and 0 otherwise A constant f'(t) ,f(t)=e^jw0t ((zero is under w))

How to solve Integration by substitution f(x)=(S) e^3x-2 dx

u=3x-2 du/dx= 3 dx=(1/3) f'(x)=(S) e^u (1/3) du f'(x)= (1/3) e^u f'(x) (1/3) e^3x-2 + K u was used as a substittute

How to evaluate integrals involving logarithms (S) x ln x dx

u=ln x du/dx= x or du/dx= 1/x v=x^2/2 f(x)=(x^2/2) ln x - (S) (x^2/2) (1/x) dx f(x)=(x^2/2)ln x - (S) (x/2) dx f(x)=(x^2/2)ln x-x^2/4 +C