Anti-differentiation — practice questions

Find $\displaystyle\int \left(8x^3 - \frac{3}{x^2} + \sqrt{x}\right)dx$.

A curve has $\dfrac{d^2y}{dx^2} = 12x - 6$. Its gradient at the point $(1, 2)$ is 4. Find the equation of the curve.

A curve passes through the origin and has gradient function $\dfrac{dy}{dx} = kx^2 - 4x$, where $k$ is a constant. The curve has a stationary point at $x = 3$. Find $k$, the equation of the curve, and the nature of both stationary points.

Find $\displaystyle\int \left(e^{4x} + \sin 2x\right)dx$.

Find $\displaystyle\int \frac{4x - 1}{2x^2 - x + 3}\,dx$, justifying your choice of method.

Find $\displaystyle\int \frac{2x^2 + 3x - 1}{x + 2}\,dx$, explaining why the logarithm rule cannot be applied directly.

Find $\displaystyle\int 2x\left(x^2 + 7\right)^5 dx$.

Find $\displaystyle\int \cos^2 3x\,dx$.

Find $\displaystyle\int \sin^5 x\,dx$, explaining why your method differs from the one used for $\displaystyle\int \sin^2 x\,dx$.