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Definite Integrals

If α=∫10 e9x+...

Question

If $α = 1 \int 0 (e^{9 x + 3 {tan}^{- 1} x}) (\frac{12 + 9 x^{2}}{1 + x^{2}}) d x$ , where ${tan}^{- 1} x$ takes only principle values, then the value of $(ln | 1 + α | - \frac{3 π}{4})$ is

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Solution

$α = 1 \int 0 (e^{9 x + 3 {tan}^{- 1} x}) (\frac{12 + 9 x^{2}}{1 + x^{2}}) d x$
Let $9 x + 3 {tan}^{- 1} x = t$
$(\frac{12 + 9 x^{2}}{1 + x^{2}}) d x = d t$
$\Rightarrow α = 9 + \frac{3 π}{4} \int 0 e^{t} d t = e^{9 + \frac{3 π}{4}} - 1$
$\Rightarrow ln | 1 + α | - \frac{3 π}{4} = 9$

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