Example 2

Tag 08FR, permanent. Cite it as [Bourbaki, Tag 08FR].

Functions of a Real Variable, chapter V, § 3. ASYMPTOTIC EXPANSIONS OF FUNCTIONS OF A REAL VARIABLE

Let $f(x) = \frac{e^x}{x^2 + 1}$ ; one can write $f(x) = \frac{e^x}{x^2} - \frac{e^x}{x^4} + o_1 \left( \frac{e^x}{x^4} \right)$ . Prop. 8 gives the expansions

\int_a^\lambda \frac{e^t}{t^2} \, dt = \frac{e^x}{x^2} + \frac{2e^x}{x^3} + \frac{6e^x}{x^4} + o_2 \left( \frac{e^x}{x^4} \right), \quad \int_a^\lambda \frac{e^t}{t^4} \, dt = \frac{e^x}{x^4} + o_3 \left( \frac{e^x}{x^4} \right)

whence, on adding,

\int_a^\lambda \frac{e^t}{t^2 + 1} \, dt = \frac{e^x}{x^2} + 2 \frac{e^x}{x^3} + 5 \frac{e^x}{x^4} + o_4 \left( \frac{e^x}{x^4} \right).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Prop. 8