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#

2 Answers
Harish Chandra Rajpoot
Jul 4, 2018

#f(x)=\sec x#

Explanation:

Given that

#\int e^x(\tan x+1)\sec x\ dx#

#=\int e^x\tan x\sec x\ dx+\int e^x\sec x\ dx#

#=e^x\int \sec x\tan x\ dx-\int (\frac{d}{dx}e^x\cdot \int \sec x\tan x\ dx)dx+\int e^x\sec x\ dx#

#=e^x\sec x-\int (e^x\sec x)dx+\int e^x\sec x\ dx#

#=e^x\sec x+C#

#=e^xf(x)+C#

we get

#f(x)=\sec x#

Notice: In general,

#\int e^x(f(x)+f'(x))\ dx#

#=\int e^xf(x)dx+\int e^x f'(x)\ dx#

#=\int e^xf'(x)dx+\int e^x f(x)\ dx#

#=e^x\int f'(x)dx-\int(\frac{d}{dx} e^x\cdot \int f'(x)\ dx)dx+\int e^x f(x)\ dx#

#=e^xf(x)-\int e^xf(x)dx+\int e^x f(x)\ dx#

#=e^xf(x)+C#

Steve M
Jul 4, 2018

# 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 #e^xf(x)+C# to get

# 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 #

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