Question
Mathematics Question on Definite Integral
If ∫sinx+1cosx−1exdx is equal to:
A
1+sinxexcosx+C
B
C−1+sinxexsinx
C
C−1+sinxex
D
C−1+sinxexcosx
Answer
1+sinxexcosx+C
Explanation
Solution
I=∫(sinx+1cosx−1)exdx
I=∫ex(sinx+1cosx−1+sinx1)dx
Let f(x)=1+sinxcosx
f′(x)=(1+sinx)2−sinx(1+sinx)−cos2x
=(1+sinx)2−1−sinx=−1+sinx1
I=∫ex(f(x)+f′(x)]dx
I=ex⋅f(x)=ex1+sinxcosx
I=1+sinxcosx⋅ex+C