∫0^π/10 (sin5x)/(1 + cos^2 5x) dx

Name: ∫0^π/10 (sin5x)/(1 + cos^2 5x) dx
Uploaded: 2022-06-06T06:06:26-08:00
Duration: 5 min 29 s
Channel: Adi S
Description: ∫0^π/10 (sin5x)/(1 + cos^2 5x) dx

Transcript

00:01 In this question we need to evaluate the integral.

00:03 So the a part is integration cos square 3x, sign 3x d x.

00:18 So here we'll start it as let cos 3x3x is equals to t.

00:28 So here on taking the derivative of it we get negative sign sign sign sign 3x .3d .3 dx is equal to d t so from here we get sign 3x dx is equals to negative 1 divided by 3 d t now we can easily replace the given integral by the terms as integration at a place of cos square 3x will put t square and at a place of of sine 3x dx will put negative 1 divided by 3 d t now we can easily solve this integral and we get first we bring out this constant term out of the integral that is negative 1 divided by 3 integration t square d t and it will be written as negative 1 divided by 3 t cubed divided by 3 plus c where c is a constant.

01:47 Now we'll substitute the value of t that we have letted above.

01:53 So we have negative 1 divided by 3, sorry 1 divided by 9, cos 3x whole cube plus c and we get the required value of the integral as negative 1 divided by 9 cos 3x whole cube plus c and we get the required value of the integral as negative cube plus c so this is the required value of the integral part b here we have the integral 5x square root 3x square plus 5 d x so here we let 3x square plus 5 dx so here we let 3x square plus 5 as p so here on differentiating this equation with respect to x we get 6x dx is equal to dp the integration of a constant term will be 0 so we get this so from here we have x d x is equal to 1 divided by 6 dp so we substitute the evaluated value in the integral and now the integral will be written as integration at a place of x d x d x d x will put 1 divided by 6 dp 5 will be as it is and at a place of 3x square plus 5 we put p which is in the square root so we have this as a integral which can be solved as integration 5 divided by 6 p is in under root dp.

04:07 Now we will bring the constant terms out that is 5 divided by 6.

04:12 It can be written as p raised to the power 1 by 2 dp...

Question

\int_0^{\pi/10} \frac{\sin 5x}{1 + \cos^2 5x} dx

Question

\int_0^{\pi/10} \frac{\sin 5x}{1 + \cos^2 5x} dx

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