What is {x^{e-1}+e^{x-1}}{x^{e}+e^{x}}dx equal to?

Correct Answer is : 1eln⁡(xe+ex)+c1/e (x^{e}+e^{x})+ce1ln(xe+ex)+c

Correct Answer is : 1eln⁡(xe+ex)+c1/e (x^{e}+e^{x})+ce1ln(xe+ex)+c

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UPSC NDA I 2017 Mathematics Solved Paper

Question : 62 of 120

Marks: +1, -0

What is

\int\frac{x^{e-1}+e^{x-1}}{x^{e}+e^{x}}dx

equal to?

Solution:

Concept:

Recognize that the numerator is almost the derivative of the denominator, allowing a substitution of the form

\int\frac{dt}{t}

Explanation:

Let

I = \int\frac{x^{e-1} + e^{x-1}}{x^e + e^x} \, dx

Set

t = x^e + e^x

Differentiate:

dt = (e x^{e-1} + e^x) \, dx

Factor

e

dt = e(x^{e-1} + e^{x-1}) \, dx

Thus,

(x^{e-1} + e^{x-1}) \, dx = \frac{dt}{e}

Substitute into

I

I = \frac{1}{e} \int\frac{dt}{t}

Integrate:

\frac{1}{e} \ln|t| + c

Replace

t

with

x^e + e^x

I = \frac{1}{e} \ln(x^e + e^x) + c

Answer:

\displaystyle \frac{1}{e} \ln (x^e + e^x) + c

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