turkey-yks 2014 Q45

turkey-yks · Other · lys1-math Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula

For every integer $m$ greater than 1
$$\int \tan ^ { m } x d x = \frac { 1 } { m - 1 } \tan ^ { m - 1 } x - \int \tan ^ { m - 2 } x d x$$
the equality is satisfied. Accordingly, what is the value of the integral $\int _ { 0 } ^ { \frac { \pi } { 4 } } \tan ^ { 4 } \mathrm { xdx }$?
A) $\frac { 2 \pi + 3 } { 4 }$
B) $\frac { 4 \pi - 3 } { 8 }$
C) $\frac { 3 \pi - 8 } { 12 }$
D) $\pi + 2$
E) $2 \pi + 1$

For every integer $m$ greater than 1

$$\int \tan ^ { m } x d x = \frac { 1 } { m - 1 } \tan ^ { m - 1 } x - \int \tan ^ { m - 2 } x d x$$

the equality is satisfied.\\
Accordingly, what is the value of the integral $\int _ { 0 } ^ { \frac { \pi } { 4 } } \tan ^ { 4 } \mathrm { xdx }$?\\
A) $\frac { 2 \pi + 3 } { 4 }$\\
B) $\frac { 4 \pi - 3 } { 8 }$\\
C) $\frac { 3 \pi - 8 } { 12 }$\\
D) $\pi + 2$\\
E) $2 \pi + 1$

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