# Why can't I integrate trigonometric functions without making a substitution?

For example, $\int\sin^3x$ is turned into $\int sinxsin^2x$ then a substitution for $\sin^2x$ is made. What I would have done was $\int(sinx)^3$ and integrate via recognition: $\int (ax+b)^n dx$ =$\frac{(ax+b)^{n+1}}{a(n+1)} + C$ . However this will give a different answer from the correct one. Why is that? Why can't this reverse chain rule work for trigonometric functions?

• How does $\sin x$ correspond to $ax+b$? – Arthur Dec 28 '16 at 18:17
• Because sine is not a linear function, so your rule for the power of a linear function won't apply. If you don't want to make a substitution, one recourse is to switch to complex exponentials. – J. M. isn't a mathematician Dec 28 '16 at 18:17
• You would do as well to use the identity $\sin^2x=1-\cos^2x$ - no substitution required – Mark Bennet Dec 28 '16 at 18:25
• You're also leaving out the "$dx$" in your notation which is absolutely critical for any substitution – MPW Dec 28 '16 at 18:34
• Apparently another person led astray by the notation $\sin x$ instead of $\sin(x)$. – Robert Israel Dec 28 '16 at 18:37

This is basically boils down to the fact that $$(x^n)' = n \, x^{n-1}$$ but $$(\sin^n x)' = n \, \sin^{n-1} x \, \cos x.$$ You can see how these are not 'analogous' so you can't generalize the integral formula for $(a \, x + b)^n$

Clearly this is not the case. For example, by your logic, we have

$$\int(x^2)^2\ dx=\frac13(x^2)^3+C=\frac13x^6+C$$

But it should be obvious that

$$\int(x^2)^2\ dx=\int x^4\ dx=\frac15x^5+C$$

If you can't do this with $x^2$, I don't see any reason why such logic would work on $\sin x$, which is far more complicated.

Particularly, you should differentiate your result to check if it is right, and don't forget to apply chain rule. For example,

$$\frac d{dx}\frac{\sin^4x}4=\cos x\sin^3x\ne\sin^3x$$

To do backwards chain rule, you needed a $\cos x$ out in the front, which there is none.

When trigonometric function have some power i.e non-linear functions. You can't do it simply. You have to use substitution or integration by parts.

$\int\sin^3x$
=$\int sinxsin^2x$
=$\int sinx (1- cos^2x)$