Why doesn't substiuting work here?

Question

I know that:$$\int \cos x dx = \sin x +C$$ Substiute $x$ for $ax+b$: $$\int \cos(ax+b) dx = \sin(ax+b) +C$$ but according to my book: $$\int \cos(ax+b) dx = \frac{1}{a}\sin(ax+b) +C$$ Why doesn't substiuting work here?

Answer 1

You need to substitute $x$ for $ax + b$ not only in argument of $\cos$, but also in differential: $\int \cos(ax + b)\, d\color{red}{(ax + b)} = \sin(ax + b)$.

This is equivalent to your citation, as $\int \cos(ax + b)\, d(ax + b) = a \cdot \int \cos(ax + b)\, dx$.