The function \(f\) is defined by \(f(x) = (4x + 2)^{-2}\) for \(x > -\frac{1}{2}\).
Find \(\int_{1}^{\infty} f(x) \, dx\).
Find \(\int \left( 4x + \frac{6}{x^2} \right) \, dx\).
Find \(\int_{1}^{\infty} \frac{1}{(3x - 2)^{\frac{3}{2}}} \, dx\).
A curve has equation \(y = \frac{1}{k} x^{\frac{1}{2}} + x^{-\frac{1}{2}} + \frac{1}{k^2}\) where \(x > 0\) and \(k\) is a positive constant.
It is given instead that \(\int_{\frac{1}{4}k^2}^{k^2} \left( \frac{1}{k} x^{\frac{1}{2}} + x^{-\frac{1}{2}} + \frac{1}{k^2} \right) \, dx = \frac{13}{12}\).
Find the value of \(k\).
Showing all necessary working, find \(\int_{1}^{4} \left( \sqrt{x} + \frac{2}{\sqrt{x}} \right) \, dx\).
Find \(\int \frac{2}{\sqrt{5x - 6}} \, dx\) and hence evaluate \(\int_{2}^{3} \frac{2}{\sqrt{5x - 6}} \, dx\).
Find \(\int (3x - 2)^5 \, dx\) and hence find the value of \(\int_0^1 (3x - 2)^5 \, dx\).
Find \(\int \left( x^3 + \frac{1}{x^3} \right) \, dx\).
Find \(\int \left( x + \frac{1}{x} \right)^2 \, dx\).
Evaluate \(\int_{0}^{1} \sqrt{3x + 1} \, dx\).
A curve is such that its gradient at a point \((x, y)\) is given by \(\frac{dy}{dx} = x - 3x^{-\frac{1}{2}}\). It is given that the curve passes through the point \((4, 1)\).
Find the equation of the curve.
The equation of a curve is such that \(\frac{dy}{dx} = 12\left(\frac{1}{2}x - 1\right)^{-4}\). It is given that the curve passes through the point \(P(6, 4)\).
(a) Find the equation of the tangent to the curve at \(P\).
(b) Find the equation of the curve.
The equation of a curve is such that \(\frac{dy}{dx} = 3(4x - 7)^{\frac{1}{2}} - 4x^{-\frac{1}{2}}\). It is given that the curve passes through the point \((4, \frac{5}{2})\).
Find the equation of the curve.
The equation of a curve is such that \(\frac{d^2y}{dx^2} = 6x^2 - \frac{4}{x^3}\). The curve has a stationary point at \((-1, \frac{9}{2})\).
(a) Determine the nature of the stationary point at \((-1, \frac{9}{2})\).
(b) Find the equation of the curve.
(c) Show that the curve has no other stationary points.
(d) A point \(A\) is moving along the curve and the \(y\)-coordinate of \(A\) is increasing at a rate of 5 units per second. Find the rate of increase of the \(x\)-coordinate of \(A\) at the point where \(x = 1\).
A curve with equation \(y = f(x)\) is such that \(f'(x) = 2x^{-\frac{1}{3}} - x^{\frac{1}{3}}\). It is given that \(f(8) = 5\).
Find \(f(x)\).
A curve is such that \(\frac{dy}{dx} = \frac{8}{(3x + 2)^2}\). The curve passes through the point \((2, 5\frac{2}{3})\).
Find the equation of the curve.
A curve with equation \(y = f(x)\) is such that \(f'(x) = 6x^2 - \frac{8}{x^2}\). It is given that the curve passes through the point \((2, 7)\).
Find \(f(x)\).
The gradient of a curve is given by \(\frac{dy}{dx} = 6(3x-5)^3 - kx^2\), where \(k\) is a constant. The curve has a stationary point at \((2, -3.5)\).
(a) Find the value of \(k\).
(b) Find the equation of the curve.
The equation of a curve is such that \(\frac{dy}{dx} = \frac{3}{x^4} + 32x^3\). 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 \(\frac{dy}{dx} = \frac{6}{(3x - 2)^3}\) and \(A(1, -3)\) lies on the curve.
Find the equation of the curve.