Question

how do you know when to try a trig substitution for an integral?

sometimes it doesn't seem obvious :/

Answer 1

square root with both square terms, if you know what i mean :)

Answer 2

you can use them when the integrand contains radical form (only second radical) of
\[\sqrt{a^2-u^2}\]\[\sqrt{a^2+u^2}\]\[\sqrt{u^2-a^2}\]

or even without the radical, just follow the pattern.

a is constant

Answer 3

when the integrand involves
1.\[a^2 -x^2\]
2.\[a^2 + x^2\]
3.\[x^2 -a^2\]
try
1. \[x=asin \theta\]
2. \[x=atan\]
3.\[x=asec \theta\]

Answer 4

2.
\[x=atan \theta\]

Answer 5

@.Sam.
yup

Answer 6

sometimes we have the second radical raised to a power, and that works to

eg
\[\int\limits{\frac{1}{(x^2+a^2)^{3/2}}}dx\]

Answer 7

so should i think of that as

\[(\sqrt{x^2+a^2})^3\]

i presume

Answer 8

will it work for any f(x^2 +a^2) ?

Answer 9

\[\int\limits{f(x^2 + a^2)}dx\]

let x = atanu

\[\int\limits{f(\sec^2u)}\sec^2u dx\]

so i guess we need f such that it cancels somehow.

in the previous comment it comes to integral(cosu du)

sinu + C

but u = arctan(x/a)

so its sin(arctan(x/a)) ?

can this be written without trig ?

Answer 10

Use right triangles.