First slide

Methods of integration

Question

$If f (x) = \int \frac{5 x^{8} + 7 x^{6}}{{(x^{2} + 1 + 2 x^{7})}^{2}} d x, and f (0) = 0, then the value$ $of f (1) is$

Moderate

A

$- 1 / 4$
B

$1 / 4$
C

$1 / 2$
D

$- 1 / 2$

Solution

$\begin{aligned} f (x) = \int \frac{5 x^{8} + 7 x^{6}}{x^{14} {(2 + \frac{1}{x^{7}} + \frac{1}{x^{5}})}^{2}} d x \\ = \int \frac{\frac{5}{x^{6}} + \frac{7}{x^{8}}}{{(2 + \frac{1}{x^{7}} + \frac{1}{x^{5}})}^{2}} d x \end{aligned}$
$Put, 2 + \frac{1}{x^{7}} + \frac{1}{x^{5}} = t$
$\begin{array}{l} \Rightarrow f (x) = - \int \frac{d t}{t^{2}} \\ = \frac{1}{t} + c \\ = (\frac{x^{7}}{2 x^{7} + x^{2} + 1}) + c \\ f (0) = 0 \\ \Rightarrow c = 0 \\ \Rightarrow f (x) = (\frac{x^{7}}{2 x^{7} + x^{2} + 1}) \\ \Rightarrow f (1) = \frac{1}{4} \end{array}$

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