Help me to understand this integration.

Consider this integral $\displaystyle I=\int { \tan { x } \sec ^{ 2 }{ x } } dx$.

I solved it in this way $let,\quad \sec { x } =s\\ then,\quad \tan { x } \sec { x } dx=ds\\ \therefore I=\int { s } ds=\cfrac { { s }^{ 2 } }{ 2 } +c=\cfrac { \sec ^{ 2 }{ x } }{ 2 } +c$

is it wrong?

Because my book had this solution: $I=\int { \tan { x } \sec ^{ 2 }{ x } } dx\\ let,\quad \tan { x } =t\\ then,\quad \sec ^{ 2 }{ x } dx=dt\\ I=\int { t } dt=\cfrac { { t }^{ 2 } }{ 2 } +c=\cfrac { \tan ^{ 2 }{ x } }{ 2 } +c$

How come?

Update!

Ok, I got it thanx Brilliant community.

#Integration #HelpMe! #Help #Confused

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Comments

Both of the approaches here are absolutely correct. It's just that the $c$ 's are different.

If the constants of integration in the first approach and the second approach are $c_1$ and $c_2$ respectively, we have $c_1+\frac{1}{2}=c_2$ .

Does this help?

Mursalin Habib - 6 years, 5 months ago

Thanx for replying.

So, beacuse of the constant will the definite integral be different?

Soumo Mukherjee - 6 years, 5 months ago

Uh no. Because the difference in the constants get cancelled out on a definite integral. :-0 o_O

So if it was something like $\int _{ a }^{ b }{ tanx{ sec }^{ 2 }x\quad dx } = \frac{ {sec}^{2}b}{2 } + { c }_{ 1 } - \frac{ {sec}^{2}a}{2 } - { c }_{ 1 }$ and similar thing will happen with $\frac{{tan}^2{x}}{2} + {c}_{2}$ .

And because the difference between $\frac{{tan}^{2}x}{2}$ and $\frac{{sec}^{2}x}{2}$ is a constant. The answer will be the same.

Vishnuram Leonardodavinci - 6 years, 5 months ago

Yo! :P If you're curious, feel free to check out Trevor's note on this topic! :D

Vishnuram Leonardodavinci - 6 years, 5 months ago

thanx for replying. :)

i will look into the note.

Soumo Mukherjee - 6 years, 5 months ago

Both answers absolutely correct since, adding 1/2 to second answer (TANx^2) and subtracting 1/2 from c won't matter...

ashay wakode - 6 years, 5 months ago

both answers are ultimately the same. from the infinite set of c if you take out 0.5 and add it to the answer given in the book, it leads to the answer you found out! As simple as that!

Wrik Bhadra - 6 years, 5 months ago

What is equal to i^2n+2

manthar Ali - 6 years, 5 months ago

If 'i' here is considered as the imaginary number then the value depends strictly on n. For n $in$ Z (the set of integers), the value will alternate between -1 and 1.

Wrik Bhadra - 6 years, 4 months ago

The solution given in the book is right

Avishkar Raut - 6 years, 5 months ago

the c in the book method will be 1/2 less than the c in your method

Rutwik Dhongde - 6 years, 5 months ago

i dont understand the formula

shweta dalal - 6 years, 4 months ago

Math	Appears 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}`	$\boxed{123}$