I=∫_0^1 dx1+x^2 ∫dx1+x^2=tan^ 1 x=F(x) Now, I=F(1) F(0) I=tan^ 1(1) tan^ 1(0) I=π4 ...

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Theorems on Integration

∫01 dx1+x2

Question

${\int_{0}}^{1} \frac{d x}{1 + x^{2}}$

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Solution

$I = {\int_{0}}^{1} \frac{d x}{1 + x^{2}}$

$\int \frac{d x}{1 + x^{2}} = {tan}^{- 1} x = F (x)$
Now,
$I = F (1) - F (0)$
$I = {tan}^{- 1} (1) - {tan}^{- 1} (0)$
$I = \frac{π}{4}$

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