Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals2 Marks
Question
Evaluate the integral $\int_{0}^{1} \frac{x}{x^{2}+1} d x$ using substitution.
✓
Answer
Given integral is: $\int_{0}^{1} \frac{x}{x^{2}+1} d x$
Let $x^2 + 1 = t$
$\Rightarrow 2xdx = dt$
⇒ xdx = $\frac{1}{2}$ dt
When x = 0, t = 1 and when x = 1, t = 2
$\Rightarrow \int_{0}^{1} \frac{x}{x^{2}+1} d x=\int_{1}^{2} \frac{d t}{2 t}$
$=\frac{1}{2} \int_{1}^{2} \frac{d t}{t}$
$=\frac{1}{2}\left[\log |t| ]_{1}^{2}\right.$
= $\frac{1}{2}[\log 2-\log 1]$
= $\frac{1}{2} \log 2$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.