Key to "Find the indefinite integral ∫sec⁡4xdx"

Minerva Kline

Answered question

2021-11-07

Find the indefinite integral

\int \sec 4 x d x

Vaing1990

Beginner2021-11-08Added 16 answers

Step 1
Having said that,

\int \sin (t) d t = - \cos (t) + C

...(1)
Given integral is what we have as

I = \int \sin (4 x) d x

Step 2
Let us consider that,
4x=t
4dx=dt

d x = \frac{1}{4} d t

Thus, integral becomes as

I = \int \frac{1}{4} \sin (t) d t

= \frac{1}{4} \int \sin (t) d t

= \frac{1}{4} (- \cos (t) + C)

(By using equation (1))

= - \frac{1}{4} \cos (t) + C

On substituting back t=4x, we get the value of required integral as

\int \sin 4 (x) d x = \frac{1}{4} (- \cos 4 (t)) + C

= - \frac{1}{4} \cos 4 (t) + C

Hence, value of integral

\int \sin (4 x) d x i s - \frac{1}{4} \cos 4 (t) + C

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