• A Level Maths

Demo

• A Level
• Rational Functions

1. Find $$\int\dfrac{3x-5}{x-3}\ \mathrm{d}x$$

$3x + 4\ln|x-3| + c$

2. Find $$\int\dfrac{6x-5}{4x^2-25}\ \mathrm{d}x$$

$\ln|2x+5| + \dfrac{1}{2}\ln|2x-5| + c$

3. Find the exact value of $$\int_1^2\dfrac{3}{9-x^2}\ \mathrm{d}x$$

$\dfrac{1}{2}\ln\dfrac{5}{2}$

4. Find $$\int\dfrac{2}{x^2-1}\ \mathrm{d}x$$

$\ln|x-1| - \ln|x+1| + c$

5. Find the binomial expansion of $$\dfrac{3+9x}{(1+x)(3+5x)}$$ up to and including the term in $x^2$, and state the range of values of $x$ for which this is valid.

$1 + \dfrac{1}{3}x-\dfrac{23}{9}x^2$ for $|x| < \dfrac{3}{5}$

6. Find $$\int\dfrac{7x-3}{(x+1)(3x-2)}\ \mathrm{d}x$$

$2\ln|x+1| + \dfrac{1}{3}\ln|3x-2| + c$

7. Find the binomial expansion of $$\dfrac{19x-3}{(1+2x)(3-4x)}$$ up to and including the term in $x^2$, and state the range of values for $x$ for which this is valid.

$-1 + 7x - \dfrac{22}{3}x^2$, $|x| < \dfrac{1}{2}$

8. By decomposing $\dfrac{3x-1}{(1-2x)^2}$ into partial fractions, find its series expansion, in ascending powers of $x$, up to and including the term in $x^3$.

$-1 -x + 4x^3$

9. Find the binomial expansion of $$\dfrac{1+4x}{(1+x)(1+3x)}$$ up to the term in $x^3$, and state the range of values of $x$ for which this expansion is valid.

$1-3x^2+12x^3$, $|x| < \dfrac{1}{3}$

10. Find the binomial expansion of $$\dfrac{3x-1}{(1-x)(2-3x)}$$ up to and including the term in $x^2$, and state the range of values of $x$ for which this is valid.

$-\dfrac{1}{2} + \dfrac{1}{4}x + \dfrac{11}{8}x^2$
$|x| < \frac{2}{3}$

11. Find $$\int\dfrac{10x^2+8}{(x+1)(5x-1)}\ \mathrm{d}x$$

$2x - 3\ln|x+1|+\dfrac{7}{5}\ln|5x-1| + c$

12. Find the exact value of $$\int_{-1}^0\dfrac{5-8x}{(2+x)(1-3x)}\ \mathrm{d}x$$

$\dfrac{11}{3}\ln2$

13. Find the exact value of $$\int_0^4\dfrac{19x-2}{(5-x)(1+6x)}\ \mathrm{d}x$$

$\dfrac{8}{3}\ln 5$

14. The gradient of a curve is given by $$\dfrac{4x^3-2x^2+16x-3}{2x^2-x+2}$$ Given that $(-1,2)$ lies on the curve, find the equation of the curve.

$y = x^2 + 3\ln(2x^2-x+2) + 1 - 3\ln 5$

15. Find $$\int\dfrac{28+4x^2}{(1+3x)(5-x)^2}\ \mathrm{d}x$$

$\dfrac{1}{3}\ln|1+3x| + \ln|5-x| + \dfrac{8}{5-x} + c$

16. Find the series expansion of $$\dfrac{3x^2+16}{(1-3x)(2+x)^2}$$ in ascending powers of $x$, up to and including the term in $x^3$.

$4 + 8x + \dfrac{111}{4}x^2 + \dfrac{161}{2}x^3$

17. $$\mathrm{f}(x) = \dfrac{27x^2+32x+16}{(1-x)(3x+2)^2}$$

1. Find the series expansion of $\mathrm{f}(x)$, in ascending powers of $x$, up to and including the term in $x^2$.
2. Find the percentage error in using this to estimate the value of $\mathrm{f}(0.2)$.

1. $4 + \dfrac{39}{4}x^2$
2. Estimate: $4.39$, exact: $\dfrac{2935}{676}$ error: $0.0111... = 1.11\%$

18. Find the series expansion of $$\dfrac{2x^2+5x-10}{(x-1)(x+2)}$$ in ascending powers of $x$, up to and including the term in $x^2$.

$5+\dfrac{3}{2}x^2$