Click on a box with a + inside to expland the section.

Substitutions

Maple can do substitutions for you. I'm not sure how valuable this is, though.

> R:=Int((sin^2)(x)*cos(x),x);

> R1:=changevar(u=sin(x),R,u);

> S:=Int((2+3*x)^3,x);

> S1:=changevar(u=2+3*x,S,u);

Integration by parts

Here are some examples of integrals which use integration by parts.

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Example 1

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> a:=x*exp(x); Define an expression called a.

> b:=int(a,x);

Example 2

> a:=x*cos(x);

> b:=int(a,x);

Example 3

> a:=x*ln(x);

> b:=int(a,x);

Example 4

> a:=(x^3)*sqrt(x^2+3);

> b:=int(a,x);

> c:=(1/3)*x^2*(x^2+3)^(3/2) -(2/15)*(x^2+3)^(5/2);

> simplify(b-c); These two are really the same

Example 5

> a:=ln(x);

> b:=int(a,x);

Example 6

> a:=(x^n)*ln(x);

> b:=int(a,x);

Partial Fractions

Here are some examples which require partial fractions.

Example 1

> a:=(x^2+2*x)/((x+1)^2);

> b:=int(a,x);

Example 2

> a:=1/(x^2+x-6);

> b:=int(a,x);

Trigonometric integrals

Note that Maple can do the integrals, but its answer is not always in the simplest form, nor does it always look like what we get using substitutions and other tricks.

> int((sin^3)(x),x);

> diff((1/3)*(cos^3)(x)-cos(x),x);

> int(cos(x)*(sin^2)(x),x);

> int((cos^3)(x)*(sin^4)(x),x);

> int((sin^4)(x),x);

> int(tan(x),x);

> int(tan(x)*sec(x)^2,x);

> int(1/sqrt(1-x^2),x);

> int((x^2)/sqrt(4-x^2),x);