Evaluate: \(\int\frac{e^x}{\sqrt{e^{2x}+4e^x+13}}\,dx\)
Q.2 (B) (i) Evaluate: \(\int\frac{e^x}{\sqrt{e^{2x}+4e^x+13}}\,dx\)
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Answer: \(\log\left|e^x+2+\sqrt{e^{2x}+4e^x+13}\right|+c\)
Put \(e^x=t\), so \(e^x\,dx=dt\).
\(I=\int\frac{dt}{\sqrt{t^2+4t+13}}=\int\frac{dt}{\sqrt{(t+2)^2+3^2}}\).
Using \(\int\frac{dx}{\sqrt{x^2+a^2}}=\log\left|x+\sqrt{x^2+a^2}\right|+c\):
\(I=\log\left|t+2+\sqrt{(t+2)^2+9}\right|+c=\log\left|e^x+2+\sqrt{e^{2x}+4e^x+13}\right|+c\).
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· 28 Oct 2026
AI-verified · Oct 2026
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