A-Level CAIE Mathematics AS 1.8 Integration Question Bank

A curve with equation y=f(x) is such that $\mathrm{f}^{\prime}(x)=6 x^{2}-\frac{8}{x^{2}}$. It is given that the curve passes through the point (2,7).

Find f(x).

A curve is such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\sqrt{ }(2 x+5)$ and (2,5) is a point on the curve. Find the equation of the curve.

The equation of a curve is such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{4}{(x-3)^{3}}$ for x>3. The curve passes through the point (4,5).

Find the equation of the curve.

A curve passes through the point (4,-6) and has an equation for which $\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{-\frac{1}{2}}-3$. Find the equation of the curve.

Find $\int\left(x^{3}+\frac{1}{x^{3}}\right) \mathrm{d} x$.

A curve is such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{8}{\sqrt{ }(4 x+1)}$. The point (2,5) lies on the curve. Find the equation of the curve.

A curve is such that its gradient at a point (x, y) is given by $\frac{\mathrm{d} y}{\mathrm{~d} x}=x-3 x^{-\frac{1}{2}}$. It is given that the curve passes through the point (4,1).

Find the equation of the curve.

The function f is such that $\mathrm{f}^{\prime}(x)=5-2 x^{2}$ and ( 3,5 ) is a point on the curve y=f(x). Find f(x).

The equation of a curve is such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{3}{x^{4}}+32 x^{3}$. It is given that the curve passes through the point $\left(\frac{1}{2}, 4\right)$.

Find the equation of the curve.

A curve is such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6}{x^{2}}$ and (2,9) is a point on the curve. Find the equation of the curve.