# Difference between revisions of "Exercise 42"

Expand the following functions as far as the term indicated:

 Question 1(i) $\ln{(\cos{x})}$ to $x^{6}$ Question 1(ii) $e^{x}\cos{x}$ to $x^{5}$ Question 1(iii) $\tan{x}$ to $x^{5}$ Question 1(iv) $e^{\sin{x}}$ to $x^{4}$ Question 1(v) $\sin^{2}{x}$ to $x^{6}$ Question 1(vi) $\sin^{-1}{x}$ to $x^{5}$ Question 1(vii) $\sqrt{1+\sin{x}}$ to $x^{3}$ Question 1(viii) $\ln{\left(1+\sin{x}\right)}$ to $x^{4}$ Question 1(ix) $\frac{1}{\sqrt{1-5x}}$ to $x^{4}$ Question 1(x) $\frac{2}{\sqrt[3]{8+3h}}$ to $h^{3}$ Question 1(xi) $\ln{\left(1+e^{x}\right)}$ to $h^{2}$ Question 1(xii) $\tan^{-1}{x}$ to $h^{5}$

Find expansions as far as the term indicated for:

 Question 1(i) $\frac{x \ln{\left(1+x\right)}}{\sqrt{1+x}}$ to $x^{4}$ Question 1(ii) $\sqrt{1-x} \sin{x}$ to $x^{4}$ Question 1(iii) $e^{x} \sin{3x}$ to $x^{4}$ Question 1(iv) $e^{x} \ln{\left(1+x\right)}$ to $x^{3}$ Question 1(v) $\sin^{2}{x}$ to $x^{6}$ Question 1(vi) $\sin^{-1}{x}$ to $x^{5}$ Question 1(vii) $\sqrt{1+\sin{x}}$ to $x^{3}$ Question 1(viii) $\ln{\left(1+\sin{x}\right)}$ to $x^{4}$ Question 1(ix) $\frac{1}{\sqrt{1-5x}}$ to $x^{4}$