Giải thích các bước giải:
Câu 7:
$I=\int^1_0x\sqrt{x^2+1}dx$
$\to I=\int^1_0\dfrac 12\sqrt{x^2+1}.2xdx$
$\to I=\int^1_0\dfrac 12(x^2+1)^{\frac{1}{2}}d(x^2+1)$
$\to I=\dfrac 12.\dfrac 23(x^2+1)^{\frac{3}{2}}|^1_0$
$\to I=\dfrac 13(x^2+1)^{\frac{3}{2}}|^1_0$
$\to I=\dfrac{2\sqrt{2}}{3}-\dfrac 13$
Câu 8:
$I=\int^2_1\dfrac{ln x}{x^3}dx$
$\to I=\int^2_1\ln x.x^{-3}dx$
$\to I=\int^2_1-\dfrac 12\ln xd(x^{-2})$
$\to I=-\dfrac 12(x^{-2}.\ln x|^2_1-\int^2_1x^{-2}d(\ln x))$
$\to I=-\dfrac 12(x^{-2}.\ln x|^2_1-\int^2_1x^{-2}.\dfrac 1xdx)$
$\to I=-\dfrac 12(x^{-2}.\ln x|^2_1-\int^2_1x^{-3}dx)$
$\to I=-\dfrac 12(x^{-2}.\ln x|^2_1+\dfrac 12x^{-2}|^2_1)$
$\to I=\dfrac{3}{16}-\dfrac{\ln 2}{8}$
Câu 9:
$J=\int^1_0dx$
$\to J=x|^1_0=1$
