"Find all the antiderivatives of the following function...." was answered by our experts

amanf

Answered question

2021-02-13

Find all the antiderivatives of the following function. Check your work by taking derivatives.

f (x) = 2 \sin x + 1

timbalemX

Skilled2021-02-15Added 108 answers

Step 1
Given:

f (x) = 2 \sin x + 1

Step 2
Explanation:

f (x) = 2 \sin x + 1

Find anti derivative of f(x)

F (x) = \int f (x) d x

F (x) = \int (2 \sin x + 1) d x

F (x) = \int 2 \sin x d x + \int d x

F (x) = 2 (- \cos x) + x + C

F (x) = - 2 \cos x + x + C

Since F(x) is an anti derivative of f(x) then we get F'(x)=f(x)
Now,

F^{'} (x) = \frac{d}{d x} (F (x))

F^{'} (x) = \frac{d}{d x} (- 2 \cos x + x + C)

F^{'} (x) = - 2 \frac{d}{d x} (\cos x) + \frac{d}{d x} (x) + \frac{d}{d x} (C)

F^{'} (x) = - 2 (- \sin x) + 1 + 0

F^{'} (x) = 2 \sin x + 1

F'(x)=f(x)
Hence, the anti derivative of

f (x) = 2 \sin x + 1

are

F (x) = - 2 \cos x + x + C

where C be an arbitrary constant

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