Using integration, prove that ∫ dx/√(x² + a²) = log|x + √(x² + a²)| + c

Question

Accepted Answer

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> [!TLDR]
> Substitute $t = x + \sqrt{x^2+a^2}$; its derivative works out to $dt = \dfrac{t}{\sqrt{x^2+a^2}}dx$, which converts the integral into the elementary form $\int \dfrac{dt}{t}$.
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> **To prove:** $\displaystyle\int \frac{dx}{\sqrt{x^2+a^2}} = \log\left|x+\sqrt{x^2+a^2}\right| + c$
>
> **Step 1 — substitution:** Let
> $$t = x + \sqrt{x^2+a^2}$$
>
> **Step 2 — differentiate $t$ w.r.t. $x$:**
> $$\frac{dt}{dx} = 1 + \frac{x}{\sqrt{x^2+a^2}} = \frac{\sqrt{x^2+a^2}+x}{\sqrt{x^2+a^2}} = \frac{t}{\sqrt{x^2+a^2}}$$
>
> **Step 3 — solve for the integrand in terms of $dt$:**
> $$dt = \frac{t}{\sqrt{x^2+a^2}}\,dx \;\Rightarrow\; \frac{dx}{\sqrt{x^2+a^2}} = \frac{dt}{t}$$
>
> **Step 4 — integrate:** …

Q.Using integration, prove that ∫ dx/√(x² + a²) = log|x + √(x² + a²)| + c

Concept understanding — Standard Integral Forms

Q.Using integration, prove that ∫ dx/√(x² + a²) = log|x + √(x² + a²)| + c

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