Indefinite integrals

If \[ \frac{d}{dx} \cos x = -\sin x \quad \quad (\text{A}) \] find \[ \int \sin x dx \]

Undo the differentiation that produced the rhs

\[ \begin{align} \int \frac{d}{dx} \cos x dx &= -\int \sin x dx \\ \end{align} \ \]

Multiply both sides by -1

\[ \begin{align} -\int \frac{d}{dx} \cos x dx &= \int \sin x dx \\ \int \sin x dx &= = -\cos x + c \end{align} \ \]

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

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Now You Try:

If \[ \frac{d}{dx} \sin x = \cos x \quad \quad (\text{A}) \] find \[ \int (-\cos x) dx \]