Using a definite integral, to create a specific recurrence relation.

Question

Hello i have the integral:

$$y_n=\int_0^1\frac{x^n}{x+5}dx$$ where $ n=1,2,3,4,....,\infty$

I need to show that the integral can be represented by the recurrence relation below;

$$y_n= \frac1n-5y_{n-1}$$

I started by doing integration by parts but still cannot find the relation. I am assuming it has to do with the integration by parts method where i am going wrong.

Answer 1

HINT:

If $$y_n=\int_0^1\frac{x^n}{x+5}dx$$

$$\int_0^1\frac{x^n}{x+5}=\int_0^1\frac{x^{n-1}(x+5-5)}{x+5}dx=\int_0^1x^{n-1}dx-5\int_0^1\frac{x^{n-1}}{x+5}dx$$ as $x+5\ne0$

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Alternatively,

$$\text{So,}y_n+5y_{n-1}$$ $$=\int_0^1\frac{x^n}{x+5}dx+5\int_0^1\frac{x^{n-1}}{x+5}dx$$ $$=\int_0^1\frac{x^{n-1}(x+5)}{x+5}dx=\int_0^1x^{n-1}dx$$ as $x+5\ne0$