Trig integration

What method should I use to integrate sinxcos^2x
It seems like it's going to get very complicated with any method

by substitution, can you find an appropriate value for u?

Took me a very long time to find the right one. Usually with questions like this first thing to do is use a trig identity to change the cos^2x into cos2x. With this one cos^2x was the substitution. Very unexpected. Any tips on how to spot this quicker?

Took me a very long time to find the right one. Usually with questions like this first thing to do is use a trig identity to change the cos^2x into cos2x. With this one cos^2x was the substitution. Very unexpected. Any tips on how to spot this quicker?

You didn't need to do any substitution. When I did A Level C4, the method of Inverse Chain Rule was in the textbooks that you can simply recognise that the integral of f'(x)f^(n)(x) dx=f^(n+1)(x)/(n+1)+(C) for n not -1 and if n=-1 then you get integral f'(x)/f(x) dx =ln(f(x))+(C) and you can easily prove those result by differentiating.

Here you had the integral of sin(x)cos^2(x) dx = - integral -sin(x)cos^2(x) dx= - integral f'(x)f^2(x) dx where f(x)=cos(x) and thus the integral is just -cos^3(x)/3+(C).

However, it interests me that the simple method of Inverse Chain Rule is so underused. At University, they'll always do the substitution wasting their time doing what should be trivial integrals based on the results even in a C4 textbook.

To add to this, it is so much easier to do many integrations if you use the Inverse Chain Rule as it informs you of what integrals can easily be done like when calculating more complicated integrals you would need to separate out those integrals where possible and also when deriving many reduction formulas.

You didn't need to do any substitution. When I did A Level C4, the method of Inverse Chain Rule was in the textbooks that you can simply recognise that the integral of f'(x)f^(n)(x) dx=f^(n+1)(x)/(n+1)+(C) for n not -1 and if n=-1 then you get integral f'(x)/f(x) dx =ln(f(x))+(C) and you can easily prove those result by differentiating.

Here you had the integral of sin(x)cos^2(x) dx = - integral -sin(x)cos^2(x) dx= - integral f'(x)f^2(x) dx where f(x)=cos(x) and thus the integral is just -cos^3(x)/3+(C).

However, it interests me that the simple method of Inverse Chain Rule is so underused. At University, they'll always do the substitution wasting their time doing what should be trivial integrals based on the results even in a C4 textbook.

To add to this, it is so much easier to do many integrations if you use the Inverse Chain Rule as it informs you of what integrals can easily be done like when calculating more complicated integrals you would need to separate out those integrals where possible and also when deriving many reduction formulas.

Here you had the integral of sin(x)cos^2(x) dx = - integral -sin(x)cos^2(x) dx= - integral f'(x)f^2(x) dx where f(x)=cos(x) and thus the integral is just -cos^3(x)/3+(C).

There is no formal technique called 'The Reverse Chain Rule', as far as I am aware. In reality, it is simply becoming adept enough with the method of substitution that one can evaluate the integral without needing to use substitution.

We cannot integrate functions except for a select few, which can be integrated using the Fundamental Theorem of Calculus. This is why the method of substitution exists, so we can transform any function into a form which can be integrated. And as above, we get the Reverse Chain Rule by just recognising how the substitution would work without having to do it.

Could you explain this a bit more please