Question

\(\int\)x⁷\(\sqrt{x^4+4}\)dx = m\(\sqrt{\left(x^4+4\right)^5}\)- n\(\sqrt{\left(x^4+4\right)^3}\). Tìm m.n

Answer 1

Lời giải:

Ta có : \(P=\int x^7\sqrt{x^4+4}dx=\int x^4\sqrt{x^4+4}(x^3dx)\)

\(=\int x^4\sqrt{x^4+4}\frac{d(x^4+4)}{4}=\frac{1}{4}\int x^4\sqrt{x^4+4}d(x^4+4)\)

Đặt \(\sqrt{x^4+4}=t\Rightarrow x^4=t^2-4\)

Khi đó:

\(P=\frac{1}{4}\int (t^2-4)td(t^2)=\frac{1}{2}\int (t^2-4)t^2dt\)

\(=\frac{1}{2}\int t^4dt-2\int t^2dt=\frac{t^5}{10}-\frac{2t^3}{3}+c\)

\(=\frac{1}{10}\sqrt{(x^4+4)^5}-\frac{2}{3}\sqrt{(x^4+4)^3}+c\)

Vậy \(m=\frac{1}{10}; n=\frac{2}{3}\Rightarrow mn=\frac{1}{15}\)