Chủ đề này có 3 trả lời

[Thuat ngu]

1/ Cho biết $\int_{0}^{3}f\left ( z \right )dz= 3, \int_{0}^{4}f\left ( x \right )dx=7$. Hãy tính $\int_{3}^{4}f\left ( t \right )dt$

2/ Tính các tích phân sau:

a, $\int_{0}^{1}x.ln\left ( x^{2} +x+1\right )dx$

b, $\int_{0}^{1}ln\left ( x^{2} +1\right )dx$

[thoai6cthcstqp]

1/ Cho biết $\int_{0}^{3}f\left ( z \right )dz= 3, \int_{0}^{4}f\left ( x \right )dx=7$. Hãy tính $\int_{3}^{4}f\left ( t \right )dt$

 

Từ điều kiện bài toán ta có: $\int_{0}^{3}f\left ( t \right )dt= 3, \int_{0}^{4}f\left ( t \right )dt=7$ do đó $\int_{3}^{4}f\left ( t \right )dt=7-3=4$

[thoai6cthcstqp]

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Hình gửi kèm

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[chanhquocnghiem]

 

2/ Tính các tích phân sau:

a, $\int_{0}^{1}x.ln\left ( x^{2} +x+1\right )dx$

$\int_{0}^{1}x\ln(x^2+x+1)dx=\left [ \frac{x^2}{2}\ln(x^2+x+1) \right ]_0^1-\int_{0}^{1}\left ( \frac{x^2}{2}.\frac{2x+1}{x^2+x+1} \right )dx$

$=\frac{\ln 3}{2}-\frac{1}{2}\int _0^1\left ( 2x-1-\frac{x-1}{x^2+x+1} \right )dx=\frac{\ln 3}{2}-\frac{1}{2}.\left [ x^2-x \right ]_0^1+\frac{1}{2}\int _0^1\frac{x-1}{x^2+x+1}dx=\frac{\ln 3}{2}+\frac{1}{2}\int_{0}^{1}\frac{x-1}{x^2+x+1}dx$

Mà $\int _0^1\frac{x-1}{x^2+x+1}dx=\frac{1}{2}\int _0^1\frac{2x+1}{x^2+x+1}dx-\frac{1}{2}\int _0^1\frac{3dx}{x^2+x+1}=\frac{\ln 3}{2}-\frac{1}{2}.\left [ 2\sqrt{3}\arctan\frac{2x+1}{\sqrt{3}} \right ]_0^1=\frac{\ln 3}{2}-\frac{\sqrt{3}\ \pi}{6}$

Vậy $\int _0^1x\ln(x^2+x+1)dx=\frac{3\ln 3}{4}-\frac{\sqrt{3}\ \pi}{12}=\frac{9\ln 3-\sqrt{3}\ \pi}{12}$.