Question: "Find the indefinite integral. ∫tan⁡x3sec2⁡xdx"

High School
Other

Linda Seales

Answered question

2021-12-11

Find the indefinite integral.
$\int \sqrt[3]{\tan x} \sec^{2} x d x$

Answer & Explanation

Jenny Sheppard

Beginner2021-12-12Added 35 answers

Step 1
We have to solve the indefinite integral:
$\int \sqrt[3]{\tan x} \sec^{2} x d x$
Solving the integral by substitution method.
Assuming,
$t = \tan x$
Differentiating,
$\frac{d t}{d x} = \frac{d \tan x}{d x}$
$= \sec^{2} x$
$d t = \sec^{2} x d x$
Step 2
Substituting above values in the given integral, we get
$\int \sqrt[3]{\tan x} \sec^{2} x d x = \int \sqrt[3]{t} d t$
$= \int t^{\frac{1}{3}} d t$
$= \frac{t^{\frac{1}{3} + 1}}{\frac{1}{3} + 1} + C$ (since $\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C$ )
$= \frac{t^{\frac{1 + 3}{3}}}{\frac{1 + 3}{3}} + C$
$= \frac{t^{\frac{4}{3}}}{\frac{4}{3}} + C$
$= \frac{3}{4} t^{\frac{4}{3}} + c$
$= \frac{3}{4} {(\tan x)}^{\frac{4}{3}} + C$
Hence, value of given indefinite integration is $\frac{3}{4} {(\tan x)}^{\frac{4}{3}} + C$ .

Elois Puryear

Beginner2021-12-13Added 30 answers

Given:

\int \sec^{2} (x) \sqrt{3} {\tan (x)} d x

Substitution

u = \tan (x) \Rightarrow \frac{d u}{d x} = \sec^{2} (x)

= \int \sqrt{3} {u} d u

Integral of a power function:

\int u^{n} d u = \frac{u^{n + 1}}{n + 1}

n = \frac{1}{3} :

= \frac{3 u^{\frac{4}{3}}}{4}

Reverse replacement

u = \tan (x) :

= \frac{3 \tan^{\frac{4}{3}} (x)}{4}

\int \sec^{2} (x) \sqrt{3} {\tan (x)} d x

= \frac{3 \tan^{\frac{4}{3}} (x)}{4} + C

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Other

Didn't find what you were looking for?