#inte^x[((sin^-1x)sqrt(1-x^2)+1)/sqrt(1-x^2)]dx# ?
2 Answers
Explanation:
We have:
#inte^x[((sin^-1x)sqrt(1-x^2)+1)/sqrt(1-x^2)]dx#
Splitting up the fraction, this becomes:
#=inte^x[sin^-1x+1/sqrt(1-x^2)]dx#
Split up the integrals:
#=inte^x(sin^-1x)dx+inte^x/sqrt(1-x^2)dx#
Attempt to integrate
Let
Thus we see that the original integral equals:
#=[e^x(sin^-1x)-inte^x/sqrt(1-x^2)dx]+inte^x/sqrt(1-x^2)dx#
The integral
#=e^x(sin^-1x)#
Add the constant of integration:
#=e^x(sin^-1x)+C#
Explanation:
There is a pattern surrounding integrals involving
Since:
#d/dxe^xf(x)=e^xf(x)+e^xf'(x)=e^x[f(x)+f'(x)]#
Then:
#inte^x[f(x)+f'(x)]dx=e^xf(x)+C#
This is the case for the given problem, which simplifies to be:
#inte^x[sin^-1x+1/sqrt(1-x^2)]dx#
If
Thus the integral equals
#=e^xf(x)+C=e^x(sin^-1x)+C#