"Evaluate the following integrals. ∫tan⁡4xsec324xdx" solved and explained

SchachtN

Answered question

2021-10-21

Evaluate the following integrals.

\int \tan 4 x \sec^{\frac{3}{2}} 4 x d x

unett

Skilled2021-10-22Added 119 answers

the given indefinite integral is:

\int \tan 4 x \sec^{\frac{3}{2}} 4 x d x

let the given integral be I.
therefore,

I = \int \tan 4 x \sec^{\frac{3}{2}} 4 x d x

let the given integral be I.
therefore,

I = \int \tan 4 x \sec^{\frac{3}{2}} 4 x d x

I = \int \tan 4 x \sec^{\frac{3}{2}} 4 x d x

= \int \tan 4 x {(\sec 4 x)}^{\frac{3}{2}} d x

= \int \tan 4 x (\sec 4 x) {(\sec 4 x)}^{\frac{1}{2}} d x

= \int {(\sec 4 x)}^{\frac{1}{2}} (\sec 4 x) \tan 4 x d x

Let

\sec 4 x = t

therefore,

4 \sec 4 x \tan 4 x d x = d t

\sec 4 x \tan 4 x d x = \frac{d t}{4}

Now substitute these values in the integral I.
Therefore

I = \int {(\sec 4 x)}^{\frac{1}{2}} (\sec 4 x) \tan 4 x d x

= \int t^{\frac{1}{2}} \frac{d t}{4}

= \frac{1}{4} \int t^{\frac{1}{2}} d t

= \frac{1}{4} (\frac{t^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}) + C

= \frac{1}{4} (\frac{t^{\frac{3}{2}}}{\frac{3}{2}}) + C

= \frac{1}{4} \times \frac{2}{3} t^{\frac{3}{2}} + C

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