Home
Class 12
MATHS
Show that int e^(x)[f(x)+f'(x)]dx=e^(x)....

Show that `int e^(x)[f(x)+f'(x)]dx=e^(x).f(x)+c`
Hence, evaluate: `int e^(x)((2+sin2x)/(1+cos2x))dx`

Text Solution

Verified by Experts

`"Let I"=int e^(x)((2+sin2x)/(1+cos 2x))dx`
`=int e^(x)((2)/(1+cos 2x)+(sin 2x)/(1+cos 2x))dx`
`=int e^(x)((2)/(2cos^(2) 2x)+(2sinxcosx)/(2cos^(2)x))dx`
`=int e^(x)(sec^(2)x+tanx)dx`
`"Put f(x)"=tan x." Then "f'(x)=sec^(2)x`
`therefore I=int e^(x)[f(x)+f'(x)]dx`
`=e^(x). f(x)+c=e^(x).tan x+c`
Promotional Banner

Topper's Solved these Questions

  • INTEGRATION

    NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise Theory Questions|6 Videos

  • INTEGRATION

    NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise Example For Practice|128 Videos

  • DIFFERENTIATION

    NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise MCQ|15 Videos

  • LINE

    NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise Multiple choice question|9 Videos

Similar Questions

Explore conceptually related problems

Evaluate: int e^(2x)((1+sin2x)/(1+cos2x))dx

Evaluate: int e^(2x)((1-sin2x)/(1-cos2x))dx

Find int e^(x)((2+sin2x)/(2cos^(2)x))dx

Evaluate: int e^(x)(f(x)+f'(x))dx=e^(x)f(x)+C

int(e^(x)-sin x)/(cos x+e^(x))dx

int e^(x)(sin^(2)x+cos^(2)x)dx

Find: int e^x sqrt(1+sin2x)/(1+ cos 2x) dx.

Prove that int_0^a f(x)dx=int_0^af(a-x)dx , hence evaluate int_0^pi(x sin x)/(1+cos^2 x)dx

Prove that : int_(0)^(a) f(x) dx = int_(0)^(a) f(a-x)dx hence evaluate : int_(0)^(pi//2) (sin x)/(sin x + cos x) dx

int sin2x . e^(cos^2x) dx