Integration and Reduction Formulae Questions and Answers

For $I_n = \int \sin^n ax\, dx$, where $n$ is a positive integer and $a$ is a constant, which reduction formula is correct?

Write $\sin^2 \theta$ in terms of $\cos 2\theta$.

Find $\int \sin^2 3x\, dx$.

Find $\int \sin^4 3x\, dx$.

Given that $y = \sinh^{-1} x$, find $\frac{dy}{dx}$.

Express $4x^2 - 4x + 5$ in the form $(ax + b)^2 + c$, where $a$, $b$ and $c$ are integers.