Best solutions to - "Evaluate the integral. tan⁡xsec3xdx"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Concepcion Hale

Answered question

2022-01-07

Evaluate the integral.

\tan x \sec^{3} x d x

Answer & Explanation

Alex Sheppard

Beginner2022-01-08Added 36 answers

Step 1
Given
Evaluate the integral.

\int \tan x \sec^{3} x d x

Step 2
Let

u = \sec (x) .

Then

d u = {(\sec (x))}^{'} d x

= \tan (x) \sec (x) d x d u

= {(\sec (x))}^{'} d x

= \tan (x) \sec (x) d x

,and we have that

\tan (x) \sec (x) d x = d u .

Therefore,

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

Apply the power rule

\int u^{n} d u = \frac{u^{n + 1}}{n + 1} (n \neq - 1)

\int u^{2} d u = \frac{u^{1 + 2}}{1 + 2} = (\frac{u^{3}}{3})

Recall that

u = \sec (x) u = \sec (x) :

\frac{u^{3}}{3} = \frac{{\sec (x)}^{3}}{3}

Therefore,

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

Add the constant of integration:

\int \tan (x) \sec^{3} (x) d x = \frac{\sec^{3} (x)}{3} + C

Pademagk71

Beginner2022-01-09Added 34 answers

\int \tan x \sec^{3} x d x = \int \sec^{2} x \tan x \sec x d x = [\begin{matrix} u = \sec x \\ d u = \tan x \sec x d x \end{matrix}]

= \int u^{2} d u = \frac{u^{3}}{3} + C = \frac{\sec^{3} x}{3} + C

Result:

\frac{\sec^{3} x}{3} + C

karton

Expert2022-01-11Added 613 answers

$Step 1 Apply u-substitution Let u = \sec^{3} (x) \Rightarrow d x = \frac{1}{3 \sec^{3} (x) \tan (x)} Thus, \int \tan (x) \sec^{3} (x) d x = \frac{1}{3} \int 1 d u = \frac{1}{3} u + C = \frac{\sec^{3} (x)}{3} + C Result: \frac{\sec^{3} (x)}{3} + C$

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