a.Ta có:$(x+1)^{2n+1}=C_{2n+1}^{2n+1}x^{2n+1}+C_{2n+1}^{2n}x^{2n}+\ldots+C_{2n+1}^1x+C_{2n+1}^0$$\Rightarrow \int\limits_0^t(x+1)^{2n+1}dx=\int\limits_0^t\left(C_{2n+1}^{2n+1}x^{2n+1}+C_{2n+1}^{2n}x^{2n}+\ldots+C_{2n+1}^1x+C_{2n+1}^0\right)dx$$\Rightarrow \frac{1}{2n+2}(t+1)^{2n+2}=\frac{C_{2n+1}^{2n+1}}{2n+2}t^{2n+2}+\frac{C_{2n+1}^{2n}}{2n+1}t^{2n+1}+\ldots+\frac{C_{2n+1}^1}{2}t^2+\frac{C_{2n+1}^0}{1}t$$\Rightarrow
\int\limits_0^x\frac{1}{2n+2}(t+1)^{2n+2}dt=\int\limits_0^x\left(\frac{C_{2n+1}^{2n+1}}{2n+2}t^{2n+2}+\frac{C_{2n+1}^{2n}}{2n+1}t^{2n+1}+\ldots+\frac{C_{2n+1}^1}{2}t^2+\frac{C_{2n+1}^0}{1}t\right)dt$$\Rightarrow \frac{1}{(2n+2)(2n+3)}(x+1)^{2n+3}=\frac{C_{2n+1}^{2n+1}}{(2n+2)(2n+3)}x^{2n+3}+\frac{C_{2n+1}^{2n}}{(2n+1)(2n+2)}x^{2n+2}+\ldots+\frac{C_{2n+1}^1}{2.3}x^3+\frac{C_{2n+1}^0}{1.2}x^2$Cho $x=1$ ta được: $A=\frac{2^{2n+3}}{(2n+2)(2n+3)}$
a.Ta có:$(x+1)^{2n+1}=C_{2n+1}^{2n+1}x^{2n+1}+C_{2n+1}^{2n}x^{2n}+\ldots+C_{2n+1}^1x+C_{2n+1}^0$Lấy nguyên hàm 2 vế ta được:$\frac{1}{2n+2}(x+1)^{2n+2}=\frac{C_{2n+1}^{2n+1}}{2n+2}x^{2n+2}+\frac{C_{2n+1}^{2n}}{2n+1}x^{2n+1}+\ldots+\frac{C_{2n+1}^1}{2}x^2+\frac{C_{2n+1}^0}{1}x$Lấy nguyên hàm 2 vế lần nữa ta có:$\frac{1}{(2n+2)(2n+3)}(x+1)^{2n+3}=\frac{C_{2n+1}^{2n+1}}{(2n+2)(2n+3)}x^{2n+3}+\frac{C_{2n+1}^{2n}}{(2n+1)(2n+2)}x^{2n+2}+\ldots+\frac{C_{2n+1}^1}{2.3}x^3+\frac{C_{2n+1}^0}{1.2}x^2$Cho $x=1$ ta được: $A=\frac{2^{2n+3}}{(2n+2)(2n+3)}$
a.Ta có:$(x+1)^{2n+1}=C_{2n+1}^{2n+1}x^{2n+1}+C_{2n+1}^{2n}x^{2n}+\ldots+C_{2n+1}^1x+C_{2n+1}^0$
$\Rightarrow \in
t\limits_0^t(x+1)^{2n
+1}dx=\int\lim
its_0^t\left(C_{2
n+1}^{2n+1}x^{2n+1}+C_{2n+1}^{2n}x^{2n}+\ldots+C_{2n+1}^1x+C_{2n+1}^0\right)dx$$\Righta
rrow \frac{1}{2n+2}(
t+1)^{2n+2}=\frac{C_{2n+1}^{2n+1}}{2n+2}
t^{2n+2}+\frac{C_{2n+1}^{2n}}{2n+1}
t^{2n+1}+\ldots+\frac{C_{2n+1}^1}{2}
t^2+\frac{C_{2n+1}^0}{1}
t$
$\Righ
tarrow
\int\limits_0^x\frac{1}{2
n+2}(t+1)^{2n+2}dt=\int\limits_0^x\l
eft(\frac{C_{2n
+1}^{2n
+1}}{2n+2}t^{2n+2}+\fra
c{C_{2n+1}^{2n}}{2n+1}t
^{2n+1}+\ldots+\fra
c{C_{2n+1}^1}{2}t^2+\frac
{C_{2n+1}^0}{1}t\right)dt$
$\Rightarrow \frac{1}{(2n+2)(2n+3)}(x+1)^{2n+3}=\frac{C_{2n+1}^{2n+1}}{(2n+2)(2n+3)}x^{2n+3}+\frac{C_{2n+1}^{2n}}{(2n+1)(2n+2)}x^{2n+2}+\ldots+\frac{C_{2n+1}^1}{2.3}x^3+\frac{C_{2n+1}^0}{1.2}x^2$Cho $x=1$ ta được: $A=\frac{2^{2n+3}}{(2n+2)(2n+3)}$