Q.2 (A) (iii) Evaluate : ∫1/x(x⁶+1)dx

Answer: 1/6log|x⁶/x⁶+1|+c I=∫x⁵/x⁶(x⁶+1)dx . Put x⁶=t , so 6x⁵dx=dt . I=1/6∫dt/t(t+1)=1/6∫(1/t-1/t+1)dt=1/6[log|t|-log|t+1|]+c =1/6log|t/t+1|+c=1/6log|x⁶/x⁶+1|+c

Evaluate : \(\int\frac{1}{x(x^6+1)}dx\)

Q.2 (A) (iii) Evaluate : \(\int\frac{1}{x(x^6+1)}dx\)

Doubtly Asked 28 Oct 2026 Board exam · March 2025 hsc 12th Save WhatsApp Report

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Answer: \(\frac{1}{6}\log\left|\frac{x^6}{x^6+1}\right|+c\)

\(I=\int\frac{x^5}{x^6(x^6+1)}dx\). Put \(x^6=t\), so \(6x^5dx=dt\).

\(I=\frac{1}{6}\int\frac{dt}{t(t+1)}=\frac{1}{6}\int\left(\frac{1}{t}-\frac{1}{t+1}\right)dt=\frac{1}{6}\left[\log|t|-\log|t+1|\right]+c\)

\(=\frac{1}{6}\log\left|\frac{t}{t+1}\right|+c=\frac{1}{6}\log\left|\frac{x^6}{x^6+1}\right|+c\)

Doubtly AI · 28 Oct 2026 AI-verified · Oct 2026 Report

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Evaluate : \(\int\frac{1}{x(x^6+1)}dx\)

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