How do you find the indefinite integral of (10e^(-2x^5))/x^6 dx?
1 Answer
Let
In answering to the specific problem, I'll try to illustrate a way one can approach a problem like this.
The two main (elementary) integration techniques are integration by substitution and integration by parts .
Integration by substitution is often used in presence of composition of functions (functions "nested" in other functions) and we want to try to get rid of this "nested" structure. This can be the case. In particular, the exponent of
And to rewrite the differential
So we get
The integral of the last function cannot be expressed in terms of a finite representation of elementary functions. This means that
Although we already got an answer, let's try by parts. Integration by parts comes from the derivative of a product of functions:
So we have to see
because it's pretty easy to compute the antiderivative of
We get
We could integrate this integral by parts again, considering
but this would introduce logarithms and won't simplify the calculations.
One can express the solution by integral functions or try to write a series expansion of it.