Calculus problem #2197

Calculus Level 2

Evaluate

$N = \displaystyle \int_{0}^{1} x^2 e^x\ dx.$

The answer is 0.71828.

3 solutions

Larry Dale
Jan 17, 2014

Integration by parts. In this case the basic rule applies. N = (1st function)(2nd function) minus integral (2nd function)d(1st function) but has to be applied twice. This is obvious since the first term is squared

okay I got it. it will be e-2.

trivendra joshi - 7 years, 4 months ago

okay I've done the same but what is the correct answer? is it not e-1 ?

trivendra joshi - 7 years, 4 months ago

Okay Trivendra bit by bit. The first step gives (x^2)(e^x) minus the int of (e^x)d(x^2) and the result of this is (x^2)(e^x) minus the int of (e^x)(2xdx) [2 is taken outside the int sign]... by parts again on (e^x)(2xdx)... 2int(e^x)(xdx)=(x)(e^x) minus the int of e^x. But of course the int of e^x is just e^x. So the final expression is; (x^2)(e^x)-(2x)(e^x)+2e^x which can be factored as; e^x(x^2-2x+2). Applying the definite int x=0 gives a result of- 2e^0=-2 while x=1 gives e^1. Final answer is e^1-2e^0=2.7183...-2=0.7183...

Larry Dale - 7 years, 4 months ago

It's $(e - 2) units^{2}$

David Kroell - 7 years, 3 months ago

Todd Richert
Mar 31, 2014

Use integration by parts twice. In the first step, u = x^2 and dv = e^x dx. In the second step, u = x and dv = e^x dx. The answer is e-2 = 0.718

Priyesh Pandey
Mar 30, 2014

expand e^x then multiply its each term by x^2 then integrate first 5 to 6 terms then add them higher terms may be neglected as the sequence converges to 0.72

Calculus problem #2197

The answer is 0.71828.

3 solutions

0 pending reports