Complex partial fractions with multiple techniques

Combine partial fractions with other integration techniques (e.g., trigonometric substitution, series, or special limits) in a multi-step problem.

4 questions · Challenging +1.7

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Use the substitution $t = \tan \frac { 1 } { 2 } x$ to show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { \frac { 1 - \cos x } { 1 + \sin x } } \mathrm {~d} x = 2 \sqrt { } 2 \int _ { 0 } ^ { 1 } \frac { t } { ( 1 + t ) \left( 1 + t ^ { 2 } \right) } \mathrm { d } t$$
Express $\frac { t } { ( 1 + t ) \left( 1 + t ^ { 2 } \right) }$ in partial fractions.
Hence find $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { \frac { 1 - \cos x } { 1 + \sin x } } \mathrm {~d} x$, expressing your answer in an exact form.

Show that $$\sum _ { r = 1 } ^ { n } \frac { 5 r + 6 } { r ^ { 3 } + r ^ { 2 } } = \frac { a } { n + 1 } + b + c \sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } }$$ where $a$, $b$ and $c$ are integers whose values are to be determined. You are given that $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }$ exists and is equal to $\frac { 1 } { 6 } \pi ^ { 2 }$.
Show that $\sum _ { r = 1 } ^ { \infty } \frac { 5 r + 6 } { r ^ { 3 } + r ^ { 2 } }$ exists and is equal to $( \pi - 1 ) ( \pi + 1 )$.

Find the values of $A , B$ and $C$ for which $\frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 } \equiv A + \frac { B x + C } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 }$.
Hence express $\frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 }$ using partial fractions.
Using your answer to part (b), determine $\int _ { 0 } ^ { 2 } \frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 } \mathrm {~d} x$ expressing your answer in the form $a + \ln b + c \pi$ where $a$ is an integer, and $b$ and $c$ are both rational.