Practice with adding, subtracting and dividing basic algebraic fractions

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Integrate various functions by rewriting them as partial fractions.

Find the integral of an improper fraction.

The derivative of $\displaystyle \frac{ax+b}{cx^2+d}$ is of the form $\displaystyle \frac{g(x)}{(cx^2+d)^2}$. Find $g(x)$.

Questions testing understanding of numerators and denominators of numerical fractions.

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Find $\displaystyle \int\frac{ax^3+ax+b}{1+x^2}\;dx$. Enter the constant of integration as $C$.

Find $\displaystyle \int\frac{ax^3+ax+b}{1+x^2}\;dx$. Enter the constant of integration as $C$.

Integrate $f(x) = ae ^ {bx} + c\sin(dx) + px^q$. Must input $C$ as the constant of integration.

Find $\displaystyle \int \frac{nx^3+mx^2+px +m}{x^2+1} \;dx$

Find $\displaystyle \int \frac{nx^3+mx^2+nx + p}{1+x^2}\;dx$. Solution involves $\arctan$.

Find $\displaystyle\int \frac{ax^3-ax+b}{1-x^2}\;dx$. Input constant of integration as $C$.

Find $\displaystyle\int \frac{ax+b}{(x+c)(x+d)}\;dx,\;a\neq 0,\;c \neq d $.

Find $\displaystyle\int \frac{ax+b}{(x+c)(x+d)}\;dx,\;a\neq 0,\;c \neq d $.

Factorise $x^2+cx+d$ into 2 distinct linear factors and then find $\displaystyle \int \frac{ax+b}{x^2+cx+d}\;dx,\;a \neq 0$ using partial fractions or otherwise.

Factorise $x^2+bx+c$ into 2 distinct linear factors and then find $\displaystyle \int \frac{a}{x^2+bx+c }\;dx$ using partial fractions or otherwise.

Find $\displaystyle \int\frac{ax^3+ax+b}{1+x^2}\;dx$. Enter the constant of integration as $C$.

$\displaystyle \int \frac{bx+c}{(ax+d)^n} dx=g(x)(ax+d)^{1-n}+C$  for a polynomial $g(x)$. Find $g(x)$.

An object moves in a straight line, acceleration given by:

$\displaystyle f(t)=\frac{a}{(1+bt)^n}$. The object starts from rest. Find its maximum speed. 

The derivative of $\displaystyle \frac{ax^2+b}{cx^2+d}$ is $\displaystyle \frac{g(x)}{(cx^2+d)^2}$. Find $g(x)$.

The derivative of  $\displaystyle \frac{ax+b}{\sqrt{cx+d}}$ is $\displaystyle \frac{g(x)}{2(cx+d)^{3/2}}$. Find $g(x)$.

The derivative of $\displaystyle \frac{ax+b}{cx^2+dx+f}$ is $\displaystyle \frac{g(x)}{(cx^2+dx+f)^2}$. Find $g(x)$.

The derivative of $\displaystyle \frac{ax+b}{cx^2+dx+f}$ is $\displaystyle \frac{g(x)}{(cx^2+dx+f)^2}$. Find $g(x)$.

Inputting ratios of algebraic expressions.

Solve for $x$: $\displaystyle \frac{px+s}{ax+b} = \frac{qx+t}{cx+d}$ with $pc=qa$.

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A question to practice simplifying fractions with the use of factorisation (for binomial and quadratic expressions).