4 Find $$ \int_{-n}^{0} \mathrm{e}^{x} an ^{-1}\left(\mathrm{e}^{x} ight) \mathrm{d} x $$ in terms of $$ n $$. Deduce the exact value of $$ \int_{-\infty}^{0} \mathrm{e}^{x} an ^{-1}\left(\mathrm{e}^{x} ight) \mathrm{dx} $$

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4 Find $$ \int_{-n}^{0} \mathrm{e}^{x} 	an ^{-1}\left(\mathrm{e}^{x}ight) \mathrm{d} x $$ in terms of $$ n $$. Deduce the exact value of $$ \int_{-\infty}^{0} \mathrm{e}^{x} 	an ^{-1}\left(\mathrm{e}^{x}ight) \mathrm{dx} $$

Spencer Ross · Accepted Answer

The integral $$ \int_{-n}^{0} e^x 	an^{-1}(e^x) \, dx $$ simplifies to $$ 	an^{-1}(e^{-n}) - e^{-n} 	an^{-1}(e^{-n}) $$. The exact value of the improper integral $$ \int_{-\infty}^{0} e^x 	an^{-1}(e^x) \, dx $$ is 0.

4 Find \( \int_{-n}^{0} \mathrm{e}^{x} \tan ^{-1}\left(\mathrm{e}^{x}\right) \mathrm{d} x \) in terms of \( n \). Deduce the exact value of \( \int_{-\infty}^{0} \mathrm{e}^{x} \tan ^{-1}\left(\mathrm{e}^{x}\right) \mathrm{dx} \)

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4 Find \( \int_{-n}^{0} \mathrm{e}^{x} \tan ^{-1}\left(\mathrm{e}^{x}\right) \mathrm{d} x \) in terms of \( n \). Deduce the exact value of \( \int_{-\infty}^{0} \mathrm{e}^{x} \tan ^{-1}\left(\mathrm{e}^{x}\right) \mathrm{dx} \)

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