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

jernplate8

Answered question

2021-02-10

Find all the antiderivatives of the following function. Check your work by taking derivatives.
$p (x) = 3 \sec^{2} x$

Alannej

Skilled2021-02-12Added 104 answers

Step 1
To find an antiderivative of the function:

p (x) = 3 \sec^{2} x

Solution:
Given function is:

p (x) = 3 \sec^{2} x

.
For finding antiderivative we need to integrate the given function.

\int 3 \sec^{2} x d x = 3 \int \sec^{2} x d x

= 3 \tan x + c

(using

\int \sec^{2} x d x = \tan x + c

)
Verification:
Let,

y = 3 \tan x + c

differentiating both sides w.r.t x we get:

\frac{d y}{d x} = \frac{d}{d x} (3 \tan x + c)

\frac{d y}{d x} = 3 (\sec^{2} x) + 0

(using,

\frac{d}{d x} (\tan x) = \sec^{2} x, \frac{d}{d x} (x^{n}) = n x^{n - 1}

)

\frac{d y}{d x} = 3 \sec^{2} x

, which is given function,

p (x) = 3 \sec^{2} x

hence, veryfied
Step 2
Result:

\int 3 \sec^{2} x d x = 3 \tan x + c

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