Question

tìm họ nguyên hàm

a) \(\int\left(e^x-2\right)^5e^xdx\)

b) \(\int e^{x^2}.4xdx\)

c) \(\int\left(x-2\right)^5xdx\)

Answer 1

a.

Đặt \(e^x-2=u\Rightarrow e^xdx=du\)

\(I=\int u^5du=\dfrac{1}{6}u^6+C=\dfrac{1}{6}\left(e^x-2\right)^6+C\)

b.

Đặt \(x^2=u\Rightarrow2xdx=du\Rightarrow4xdx=2du\)

\(I=\int e^u.\left(2du\right)=2\int e^udu=2e^u+C=2.e^{x^2}+C\)

c.

Đặt \(x-2=u\Rightarrow x=u+2\) ; \(dx=du\)

\(I=\int u^5.\left(u+2\right)du=\int\left(u^6+2u^5\right)du=\dfrac{1}{7}u^7+\dfrac{1}{3}u^6+C\)

\(=\dfrac{1}{7}\left(x-2\right)^7+\dfrac{1}{3}\left(x-2\right)^6+C\)