# Integrate this monster

$\large \int \sqrt{\frac{\sec^{3}(x)}{1+\sin(x)}} \, dx =\ ?$

Note by Majed Musleh
3 years, 3 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

## Comments

Sort by:

Top Newest

It's equals to $$\displaystyle \int \sqrt{\frac1{\cos^3(x) (1+\sin(x))}} \, dx$$. Let $$y = \frac \pi2 - x$$, then it becomes

$-\int \frac1{\sqrt{\sin^3(y)(1+\cos(y))}} \, dy$

Apply Tangent half-angle substitution, then it equals to

$- \int \sqrt{\frac{1}{\left(\frac{2t}{1+t^2}\right)^3\left(1+ \frac{1-t^2}{1+t^2}\right)} }\cdot \frac{2 dt}{t^2+1} = -\frac12 \int \frac{1+t^2}{t^{3/2}} \, dt$

Which can be easily integrated from here, back substitute everything and you're done.

- 3 years, 3 months ago

Log in to reply

- 3 years, 3 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...