Answer
$\dfrac{\cos^7 x}{7}-\dfrac{\cos^5 x}{5}+c$
Work Step by Step
Consider the integral $\int \sin^3 x \cos^4 x dx$
The given integral can be re-written as:
$\int \sin^3 x \cos^4 x dx=\int \sin^2 x \cos^4 x (\sin x dx)=\int (1-cos^2 x) \cos^4 x (\sin x dx)$
Plug in $a= \cos x \implies da= -\sin x dx$
This implies that $\int (a^2-1) a^4 da =\int a^6-a^4 da$
We use the formula: $\int x^n dx=\int\dfrac{x^{n+1}}{n+1} dx$
$\dfrac{a^7}{7}-\dfrac{a^5}{5}+c=\dfrac{\cos^7 x}{7}-\dfrac{\cos^5 x}{5}+c$
