Evaluate the Integral

Problem

(∫_0^1)(e(−x2)*d(x))

Solution

1. Identify the integral as a definite integral of the Gaussian function, which does not have an elementary antiderivative.

2. Represent the integrand e(−x2) using its Taylor series expansion centered at x=0

eu=(∑_n=0^∞)((un)/(n!))

3. Substitute u=−x2 into the power series to find the expansion for the integrand.

e(−x2)=(∑_n=0^∞)(((−x2)n)/(n!))

e(−x2)=(∑_n=0^∞)(((−1)n*x(2*n))/(n!))

4. Integrate the power series term by term with respect to x from 0 to 1

(∫_0^1)((∑_n=0^∞)(((−1)n*x(2*n))/(n!))*d(x))=(∑_n=0^∞)(((−1)n)/(n!))*(∫_0^1)(x(2*n)*d(x))

5. Evaluate the definite integral of the power term.

(∫_0^1)(x(2*n)*d(x))=[(x(2*n+1))/(2*n+1)]10

(∫_0^1)(x(2*n)*d(x))=1/(2*n+1)

6. Combine the results into an infinite series representation of the value.

(∑_n=0^∞)(((−1)n)/(n!(2*n+1)))

7. Relate the integral to the error function erf(x) which is defined as 2/√(,π)*(∫_0^x)(e(−t2)*d(t))

(∫_0^1)(e(−x2)*d(x))=√(,π)/2*erf(1)

Final Answer

(∫_0^1)(e(−x2)*d(x))=(∑_n=0^∞)(((−1)n)/(n!(2*n+1)))=√(,π)/2*erf(1)

Want more problems? Check here!