Question

Calcule a seguinte integral definida:
$\int\limits^1_0 {\frac{x}{1+x^{4} } } \, dx$

Answer 1

Resposta:

Explicação passo-a-passo:

$\int\limits^1_0 {\frac{x}{1+x^4} } \, dx \\\int\limits {{\frac{x}{1+x^4} } } \, dx = \frac{1}{2} tan^{-1} (x^{2} )\\=tan^{-1}x^{2} \}^1_0 =\\ \lim_{x \to \ 1} \frac{1}{2} tan^{-1} (x)^2 = \frac{1}{2} *\frac{\pi }{4} = \frac{\pi }{8} \\\\ \lim_{x \to \ 0} \frac{1}{2} tan^{-1} (x)^2 =0$

Logo:

$\int\limits^1_0 {\frac{x}{1+x^4} } \, dx = \frac{\pi }{8}+ c$