Given #int e^x(tanx + 1 )secx dx = e^xf(x)+C#.Write f(x)satisfying above.How can you solve it ?
Please explain how can i get f(x)
#int e^x(tanx + 1 )secx dx = e^xf(x)+C#
Please explain how can i get f(x)
2 Answers
Explanation:
Given that
we get
Notice: In general,
# f(x)=secx #
Explanation:
As we are given a suggested solution then the simpler approach is to differentiate that solution and compare. So we use the product rule to differentiate
# d/dx (e^xf(x)+C) = e^x(d/dxf(x)) + (d/dxe^x)f(x) #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = e^xf'(x) + e^xf(x) #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = e^x(f'(x) + f(x)) #
And if we compare with the integrand, we have:
# e^x(f'(x) + f(x)) = e^x(tanx+1)secx#
# :. f'(x) + f(x) = (tanx+1)secx#
# :. f'(x) + f(x) = secxtanx+secx#
And by observation, we note that:
# d/dxsecx=secxtanx => f(x)=secx #