Question

Cum se rezolva aceasta integrala ? $\int\limits^2_1 { \frac{1}{x( x^{10} +1)} \, dx$

Answer 1

Salut,

Soluția de mai jos se poate vedea corect numai folosind un computer, nu un telefon, sau tabletă, etc.

$\int\limits_1^2\dfrac{1}{x(x^{10}+1)}dx=\int\limits_1^2\dfrac{x^{10}+1-x^{10}}{x(x^{10}+1)}dx=\int\limits_1^2\dfrac{x^{10}+1}{x(x^{10}+1)}dx+\int\limits_1^2\dfrac{-x^{10}}{x(x^{10}+1)}dx=\\\\=\int\limits_1^2\dfrac{1}{x}dx-\int\limits_1^2\dfrac{x^{9}}{x^{10}+1}dx=lnx\Bigg|_1^2-\dfrac{1}{10}\int\limits_1^2\dfrac{10x^{9}}{x^{10}+1}dx=ln2-\\\\-\dfrac1{10}\int\limits_1^2[ln(x^{10}+1)]^{'}dx=ln2-\dfrac{ln(x^{10}+1)}{10}\Bigg|_1^2=ln2-\dfrac{1}{10}[ln(1025)-ln2]=\\\\=ln2-\dfrac{1}{10}ln\dfrac{1025}2=\dfrac1{10}\left(10ln2-ln\dfrac{1025}2\right)=\dfrac1{10}\left[ln(1024)-ln\dfrac{1025}2\right]=\\\\\\=\dfrac1{10}ln\left(\dfrac{2048}{1025}\right).$

Green eyes.