What is ∫{dx}{x(x^{7}+1)} equal to?

Correct Answer is : 17In∣x7x7+1∣+c1/7In | {x^{7}}{x^{7}+1} | +c71In∣x7+1x7∣+c

1/2 In | {x^{7}-1}{x^{7}+1} | +c

Correct Answer is : 17In∣x7x7+1∣+c1/7In | {x^{7}}{x^{7}+1} | +c71In∣x7+1x7∣+c

Test Index

UPSC NDA I 2017 Mathematics Solved Paper

Question : 59 of 120

Marks: +1, -0

What is

∫\frac{dx}{x(x^{7}+1)}

equal to?

Solution:

Concept:

Use substitution by rewriting the integrand to simplify it.

Explanation:

Let

I = \int \frac{dx}{x(x^{7}+1)}

Multiply numerator and denominator by

x^{7}

I = \int \frac{x^{7} dx}{x^{8}(x^{7}+1)} = \int \frac{dx}{x^{8}(1 + \frac{1}{x^{7}})}

Set

t = 1 + \frac{1}{x^{7}}

Differentiate:

dt = -\frac{7}{x^{8}} dx

⇒

\frac{dx}{x^{8}} = -\frac{dt}{7}

Substitute into

I

I = -\frac{1}{7} \int \frac{dt}{t}

Integrate:

I = -\frac{1}{7} \ln|t| + C

Replace

t

I = -\frac{1}{7} \ln\left|1 + \frac{1}{x^{7}}\right| + C = -\frac{1}{7} \ln\left|\frac{x^{7}+1}{x^{7}}\right| + C

Use the logarithm property

\ln\left(\frac{a}{b}\right) = -\ln\left(\frac{b}{a}\right)

to rewrite:

I = \frac{1}{7} \ln\left|\frac{x^{7}}{x^{7}+1}\right| + C

Answer:

\frac{1}{7} \ln\left|\frac{x^{7}}{x^{7}+1}\right| + C

, which matches option D.

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