## Integrate(f(x), x, [a], [b])

Finds the indefinite integral of the given function.

If parameters a and b are given then Integrate returns the definite integral. You can also use NIntegrate for definite integrals.

## Examples

Integrate(x^2+3x-6, x)
Integrate(exp(x)*sin(x), x)
Integrate(ln(x)+1/x, x)
Integrate((x+1)/(x^2+1), x)

## Simple Integrals

Integrate(1/(a*x^2+b*x), x)
Integrate(1/cos(x), x)
Integrate(1/tan(x), x)
Integrate(sin(x)^2*cos(x), x)
Integrate(sin(3x)/sin(x), x)

## Integrals with trigonometric functions

Integrate(-3/x^2+sin(x)/(1-sin(x)), x)
Integrate((1+2cos(x))/(1+2sin(x)), x)
Integrate(cos(a*x+b)/x^3, x)
Integrate(2/sin(2x), x)
Integrate(sin(x)/sqrt(1+cos(x)), x)
Integrate(2sin(3x)/sqrt(5+cos(3x))^3, x)

## Integrals with reciprocals

Integrate(1/(1+sqrt(x)), x)
Integrate(sqrt(x^2-y), x)
Integrate(sqrt(x)/sqrt(x-2), x)
Integrate(sqrt(2x)/sqrt(x-1), x)
Integrate(sqrt(2x-1)/sqrt(x+2), x)
Integrate((3x+2)/sqrt(x-9), x)
Integrate(x^(1/4)/(1+sqrt(x)), x)
Integrate(sqrt(x)/(sqrt(x)+sqrt(x+1)), x)
Integrate((1+x^4)/(1+x^3), x)
Integrate(1/(1+x^3), x)
Integrate(1/(a*x^2+b)^2, x)

## Integrals with polynomials

Integrate(1/sqrt(1+sqrt(x)), x)
Integrate(1/(x^2*sqrt(4+x^2)), x)
Integrate(1/(x^4+4), x)
Integrate(x/(x^2+2x+5), x)
Integrate(x^2/(x^2+3x+1), x)
Integrate(x^3/(7x^2+2x-3), x)
Integrate(x^4*(x+1)^(5/3), x)

## Integrals with logarithmic functions

Integrate(ln(2*x+1), x)
Integrate((x^2+3x)*ln(x), x)
Integrate(ln(x)/(x+3)^2, x)
Integrate(sin(x)*ln(cos(x)), x)
Integrate(ln(x)/sqrt(x+1), x)
Integrate(ln(x^2+3)/(x+1)^2, x)
Integrate(x^4*ln(x)^(5/2), x)
Integrate(x^3/ln(x)^(5/2), x)

## Integrals with exponential functions

Integrate(exp(x)/((exp(x)-1)*(exp(x)+3)), x)
Integrate(exp(3x)/(1+exp(2x)), x)
Integrate(exp(-2x)/(1+exp(-2x)), x)
Integrate(1/(1+exp(2x)), x)
Integrate(exp(5x)/(1+exp(2x)), x)
Integrate(exp(x+a)/(1+exp(2x+1)), x)
Integrate(sqrt(1+exp(3x)), x)
Integrate(1/sqrt(1+exp(3x)), x)
Integrate(exp(-3*x^2), x)

## Integrals with inverse trigonometric functions

Integrate(asin(2x-1), x)
Integrate(x*asinh(x-7), x)
Integrate(x^2*asin(3x), x)
Integrate(x^3*atanh(x-1), x)
Integrate(x^4*atan(x), x)
Integrate((x+1/x^2)*atan(x), x)
Integrate(x*asin(x-2), x)

## Integrals with forced substitution

Integrate(sqrt(2+sqrt(4+sqrt(x))), x, u=2+sqrt(4+sqrt(x)))
Integrate(sqrt(2+sqrt(1+sqrt(x))), x, u=2+sqrt(1+sqrt(x)))
Integrate(sqrt(2+sqrt(4+sqrt(x))), x, u=2+sqrt(4+sqrt(x)))
Integrate((1+sqrt(x-3))^(1/3), x, u=sqrt(x-3))
Integrate(ln(x+1)^2*x, x, u=x+1)
Integrate(asin(sqrt(x)), x, u=sqrt(x), 1)

## References

http://mathworld.wolfram.com/IndefiniteIntegral.html
http://en.wikipedia.org/wiki/Integral

NIntegrate