could you help me with part a)
- Thread starter Sonal7
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We are given:
[MATH]I_n=\int_0^1 x^ne^{2x}\,dx[/MATH]
Using IBP, let's let:
[MATH]u=x^n\implies du=nx^{n-1}\,dx[/MATH]
[MATH]dv=e^{2x}\,dx\implies v=\frac{1}{2}e^{2x}[/MATH]
Hence:
[MATH]I_n=\left[\frac{1}{2}x^ne^{2x}\right]_0^1-\frac{n}{2}\int_0^1 x^{n-1}e^{2x}\,dx[/MATH]
[MATH]I_n=\frac{1}{2}e^{2}-\frac{n}{2}I_{n-1}\quad\checkmark[/MATH]
Next, we may write:
[MATH]I_0=\int_0^1 e^{2x}\,dx=\left[\frac{1}{2}e^{2x}\right]_0^1=?[/MATH]
[MATH]I_n=\int_0^1 x^ne^{2x}\,dx[/MATH]
Using IBP, let's let:
[MATH]u=x^n\implies du=nx^{n-1}\,dx[/MATH]
[MATH]dv=e^{2x}\,dx\implies v=\frac{1}{2}e^{2x}[/MATH]
Hence:
[MATH]I_n=\left[\frac{1}{2}x^ne^{2x}\right]_0^1-\frac{n}{2}\int_0^1 x^{n-1}e^{2x}\,dx[/MATH]
[MATH]I_n=\frac{1}{2}e^{2}-\frac{n}{2}I_{n-1}\quad\checkmark[/MATH]
Next, we may write:
[MATH]I_0=\int_0^1 e^{2x}\,dx=\left[\frac{1}{2}e^{2x}\right]_0^1=?[/MATH]
Oh yes, you cant use the formula for I0 as you don t know the term before than. n has to be greater than equal to 1 to use that one! I guess thats why its not working. I am sorry to ask such a question. I thought we might be able to use the formula as n =0 with negate the need for the other term. Wrong I am. Thank you so much.