Question

Help, I need the integral of: sen(x)cos(x)square root of (1+cos^2(x)) dx

Answer 1

\[\int\limits_{}^{}\sin(x)\cos(x) \sqrt{1+\cos^2(x)} dx \]

Answer 2

try a sub
try subing the inside of the square root

Answer 3

let u=1+cos^2(x)

Answer 4

how do you derive that?

Answer 5

you mean differentiate that?

Answer 6

well you need to know constant rule
and chain rule

Answer 7

\[\frac{du}{dx}=\frac{d}{dx}[1+\cos^2(x)] \\ \frac{du}{dx}=\frac{d(1)}{dx}+\frac{d [\cos(x)]^2}{dx}\]

Answer 8

i assume you know that constant rule says the derivative of a constant is 0?

Answer 9

\[\frac{du}{dx}=0+\frac{d[\cos(x)]^2}{dx}\]
Now for that last part just apply chain rule.

Answer 10

and you can do that if you know what (x^2)' equals
and what (cos(x))'=?

Answer 11

do you know what (x^2)' equals and what (cos(x))' equals?

Answer 12

no...

Answer 13

\[\frac{d}{dx}(f(g(x))=f'(x)|_{x=g(x)} \cdot g'(x) \text{ is chain rule by the way}\]

Answer 14

So you ark to integrate before you can even differentiate x^2 or cos(x)?

Answer 15

That seems unlikely to me.

Answer 16

\[\frac{d(x^n)}{dx}=nx^{n-1} \text{ is the power rule} \]

Answer 17

try to apply this rule to find the derivative of x^2

Answer 18

yes I know how to derivate, but I really dont know how to derivate cos^2(x).
that derivate would be -2cos(x)sen(x)?

Answer 19

that's it

Answer 20

\[\frac{d}{dx}(f(g(x))=f'(x)|_{x=g(x)} \cdot g'(x) \\ \frac{d}{dx}[\cos(x)]^2=2x|_{x=\cos(x)} \cdot (-\sin(x)) \\ \frac{d}{dx}[\cos(x)]^2=2\cos(x)(-\sin(x))=-2\cos(x)\sin(x)\]

Answer 21

\[\frac{-1}{2}du=\sin(x)\cos(x) dx \]

Answer 22

thak you