When \((1 + 2x)(1 + ax)^{\frac{2}{3}}\), where \(a\) is a constant, is expanded in ascending powers of \(x\), the coefficient of the term in \(x\) is zero.
(i) Find the value of \(a\).
(ii) When \(a\) has this value, find the term in \(x^3\) in the expansion of \((1 + 2x)(1 + ax)^{\frac{2}{3}}\), simplifying the coefficient.
Expand \((1 + x) \sqrt{(1 - 2x)}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients.
Expand \((2 + 3x)^{-2}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients.
(i) Simplify \((\sqrt{1+x} + \sqrt{1-x})(\sqrt{1+x} - \sqrt{1-x})\), showing your working, and deduce that
\(\frac{1}{\sqrt{1+x} + \sqrt{1-x}} = \frac{\sqrt{1+x} - \sqrt{1-x}}{2x}.\)
(ii) Using this result, or otherwise, obtain the expansion of
\(\frac{1}{\sqrt{1+x} + \sqrt{1-x}}\)
in ascending powers of \(x\), up to and including the term in \(x^2\).
(a) Expand \((2 - x^2)^{-2}\) in ascending powers of \(x\), up to and including the term in \(x^4\), simplifying the coefficients.
(b) State the set of values of \(x\) for which the expansion is valid.
Expand \((1 + 4x)^{-\frac{1}{2}}\) in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying the coefficients.
Expand \(\frac{1}{(2+x)^3}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients.
Expand \((2 + x^2)^{-2}\) in ascending powers of \(x\), up to and including the term in \(x^4\), simplifying the coefficients.
Expand \((1 - 3x)^{-\frac{1}{3}}\) in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying the coefficients.
When \((a + bx)\sqrt{1 + 4x}\), where \(a\) and \(b\) are constants, is expanded in ascending powers of \(x\), the coefficients of \(x\) and \(x^2\) are 3 and -6 respectively.
Find the values of \(a\) and \(b\).
Expand \((1 + 3x)^{\frac{2}{3}}\) in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying the coefficients.
(a) Expand \(\sqrt[3]{1 + 6x}\) in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying the coefficients.
(b) State the set of values of \(x\) for which the expansion is valid.
(a) Expand \((2 - 3x)^{-2}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients.
(b) State the set of values of \(x\) for which the expansion is valid.
Find the coefficient of \(x^3\) in the expansion of \((3-x)(1+3x)^{\frac{1}{3}}\) in ascending powers of \(x\).
Expand \(\frac{4}{\sqrt{(4 - 3x)}}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients.
Express \(\frac{4x^2 - 13x + 13}{(2x - 1)(x - 3)}\) in partial fractions.
Let \(f(x) \equiv \frac{x^2 + 3x + 3}{(x+1)(x+3)}\).
(i) Express \(f(x)\) in partial fractions.
(ii) Hence show that \(\int_0^3 f(x) \, dx = 3 - \frac{1}{2} \ln 2\).
Let \(f(x) = \frac{x^3 - x - 2}{(x-1)(x^2+1)}\).
(i) Express \(f(x)\) in the form \(A + \frac{B}{x-1} + \frac{Cx+D}{x^2+1}\), where \(A, B, C\) and \(D\) are constants.
(ii) Hence show that \(\int_2^3 f(x) \, dx = 1\).
Let \(f(x) = \frac{4x^2 + 9x - 8}{(x+2)(2x-1)}\).
(i) Express \(f(x)\) in the form \(A + \frac{B}{x+2} + \frac{C}{2x-1}\).
(ii) Hence show that \(\int_1^4 f(x) \, dx = 6 + \frac{1}{2} \ln \left( \frac{16}{7} \right)\).
Show that \(\int_1^2 \frac{u-1}{u+1} \, du = 1 + \ln \frac{4}{9}\).