Quick questions on Antiderivatives of exponential and trigonometric functions (TCE Mathematics Methods, Tasmania)

2short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is checking by differentiating back?

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The reliable way to confirm any antiderivative is to differentiate your answer and check you recover the original integrand. For instance, differentiating

-\tfrac{1}{3}\cos(3x)

gives

-\tfrac{1}{3}\cdot(-3\sin(3x)) = \sin(3x)

, which confirms the antiderivative of

\sin(3x)

What are linear inner functions?

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Each rule extends to a linear inner function

kx + c

, because the chain rule only contributes the constant factor

k

. So $

\int e^{kx + c}\,dx = \frac{1}{k}e^{kx + c} + C, \qquad \int \sin(kx + c)\,dx = -\frac{1}{k}\cos(kx + c) + C,

and similarly for cosine. The shift

does not change the reciprocal factor; only the coefficient of

matters. For example,

\displaystyle\int \cos(2x - 1)\,dx = \frac{1}{2}\sin(2x - 1) + C$.

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