7.1 Workshop Exercises 1

For more worked examples, see Gilbert and Jordan Guide to Mathematical Methods, pages 153–164.

True or False?

i) If a real polynomial of degree $n$ has $s$ real roots, then $(n-s)$ is even.

ii) The degree of a real polynomial is equal to the number of distinct roots.

iii) The quadratic polynomial $-2x^{2}+6x-7$ is irreducible over the real numbers.

iv) If $x=\sin t$ and $0\leq t\leq\pi$ then $\sqrt{1-x^{2}}=\cos t$.

See also Exercises W5.9-5.14 of MATH101. Try not to spend the whole workshop on partial fractions, for which there are lots of examples in Exercises W1.1-1.6. In particular, you should look at W1.7-1.10 on integration by substitution.

W1.1.

Express the following rational functions as the sum of a polynomial and a term $\frac{f(x)}{g(x)}$ where $\deg f<\deg g$:

W1.2.

Express the following rational functions as partial fractions, plus a polynomial if necessary:

W1.3.

Integrate the following functions:

W1.4.

a) Find the indefinite integral of the rational functions in W1.2(i)-(iv).

b) Find the indefinite integral of the rational function in W1.2(v), using the substitution $u=x^{2}+4x+6$ to integrate the part of the form $\frac{Ax+B}{x^{2}+4x+6}$.

W1.5. Using partial fractions, find the following indefinite integrals:

W1.6. i) Given that $1+3{\rm i}$ is a root of the following equation, find all of the roots:

ii) Factorize $f(x)=x^{4}-6x^{3}+26x^{2}-56x+80$ as a product of irreducible real quadratic polynomials.

iii) Use partial fractions to express $\frac{26}{f(x)}$ as a sum of expressions of the form $\frac{\alpha x+\beta}{Q(x)}$ where $Q(x)$ is an irreducible quadratic polynomial.

iv) To integrate a rational function of the form $\frac{\alpha x+\beta}{Q(x)}$ where $Q(x)=x^{2}+2bx+c$ is an irreducible quadratic, first substitute $s=x+b$ so that $Q(x)=s^{2}+(c-b^{2})$ where $c-b^{2}>0$. The resulting rational function in $s$ can be integrated using the method which is explained in 1.16-17 in the notes. (See also the examples in 1.21-24.)

Using the result from (iii), determine the indefinite integral $\int\!\frac{26}{f(x)}\,dx$.

W1.7. By using the substitution $t=\tan\frac{x}{2}$, find the definite integral

W1.8. Choose substitutions to determine the following integrals:

W1.9. Use suitable substitutions to find the integrals:

where $1<a<b$ in part (v).

W1.10. Use the substitution $x-2=2\sinh t$ to show that

recall that $\sinh^{-1}y=\log(y+\sqrt{y^{2}+1}).$