Đáp án: $\displaystyle\int^5_1x(x+1)^9dx=(\dfrac{1}{11}\cdot 6^{11}-\dfrac{1}{10}\cdot 6^{10})-(\dfrac{1}{11}\cdot 2^{11}-\dfrac{1}{10}\cdot 2^{10})$
Giải thích các bước giải:
Ta có:
$I=\displaystyle\int x(x+1)^9dx$
$\to I=\displaystyle\int (x+1)(x+1)^9-(x+1)^9dx$
$\to I=\displaystyle\int (x+1)^{10}-(x+1)^9dx$
$\to I=\displaystyle\int (x+1)^{10}-(x+1)^9d(x+1)$
$\to I=\dfrac{1}{11}\cdot (x+1)^{11}-\dfrac{1}{10}\cdot (x+1)^{10}+C$
$\to \displaystyle\int^5_1x(x+1)^9dx=(\dfrac{1}{11}\cdot 6^{11}-\dfrac{1}{10}\cdot 6^{10})-(\dfrac{1}{11}\cdot 2^{11}-\dfrac{1}{10}\cdot 2^{10})$
