First slide

Methods of integration

Question

$If f (x) = \int {(7^{x} + x^{7})}^{- 1} (7^{x} \log 7 + 7 x^{6}) d x then f (1) - f (0)$

Moderate

A

3
B

3 log 2
C

2 log 3
D

2

Solution

$f (x) = \int \frac{7^{x} \log 7 + 7 x^{6}}{x^{7} + 7^{x}} d x$
$\begin{array}{l} Let 7^{x} + x^{7} = t \\ \Rightarrow (7^{x} \log 7 + 7 x^{6}) d x = d t \\ f (x) = \int_{}^{} \frac{1}{t} d t = \log t + c \\ f (x) = \log (x^{7} + 7^{x}) + c \end{array}$
$f (1) - f (0) = \log (8) - 0 = 3 \log 2$

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