Indefinite Integration

Power Functions

100

int(6x^2-10x+2)dx

2x^3-5x^2+2x+c

200

int((x^3-2x^2)/x^2)dx

(x^2)/2-2x+c

300

int(x^3-1/sqrtx+4x)dx

x^4/4-2x^(1/2)+2x^2+c

400

int((x^4-2x^2+3)/x^2)dx

(x^3)/3-2x-3/x+c

500

The velocity of a particle is the rate of change of its position, x, at time t.  Determine an expression for the position of a particle at time, t, when:

v=3t^2-4t+18

x=t^3-2t^2+18t+c

Linear Binomials

100

int(3x-1)^3dx

(3x-1)^4/12+c

200

int(1-5x)^(3/4)dx

(-4(1-5x)^(7/4))/35+c

300

int10/(5x-2)^2dx

-2/(5x-2)+c

400

int7/3(2x-1)^6dx

(2x-1)^7/6+c

500

Determine the equation of the curve that passes through the point (2,3) and has a gradient given by:

dy/dx=(3x-8)^2

y=1/9(3x-8)^3+3 8/9

Exponential Functions

100

int2e^(3x)dx

(2e^(3x))/3+c

200

int9e^(-3x)dx

-3e^(-3x)+c

300

int(10e^(-2x)-2e^(-3x)+2x)dx

-5e^(-2x)+(2e^(-3x))/3+x^2+c

400

int(e^x(2e^(3x)-2))/e^(2x)dx

e^(2x)+2/e^x+c

500

Determine the equation of the function that passes through the point (0,3) and has a gradient function of 

f'(x)=e^(2x)-1/e^(2x)

f(x)=e^(2x)/2+1/(2e^(2x))+2

Trigonometric Functions

100

int9sin(-3x)dx

3cos(-3x)+c

200

int-3cos((3x)/2)dx

-2sin((3x)/2)+c

300

int3sin(2x-3)dx

-3/2cos(2x-3)+c

400

int(2cos((3x)/2)-3sin(4x))dx

4/3sin((3x)/2)+(3cos(4x))/4+c

500

Determine the equation of the function that has the gradient function below and passes through the point indicated:

f'(x)=2cos(3x-pi/2) and (pi/2,-1)

f(x)=2/3sin(3x-pi/2)-1

Linear Reciprocal Functions

100

int5/xdx

5 log_e lxl + c

200

int(-7)/xdx, x>0

-7ln(x)+c

300

int(4/(2x-1))dx

2 log_e l2x-1l + c

400

int((3x^2+2)/x)dx, x>0

(3x^2)/2+2lnx+c

500

Find the equation of the curve that passes through the point (e,0) and has the gradient function:

f'(x)=2x+7/x, x>0

f(x)=x^2+7ln(x)-e^2-7