Can you please help me with the following integrals?

Question

1- integral [x tan (x) dx]
2-integral [1/(sinx+cosx) dx]

Thanks in advance

Answer 1

Integral no. 1 is very complicated, it involves polylogarithmic identities with complex/imaginary numbers.

For the integral no. 2, use the following identities:

sin(x) = cos( (pi/2) - x )

cos(A) + cos(B) = 2 * cos{ (A + B) / 2 } * cos{ (A - B) / 2 }

The answer is:

integral [1/(sinx+cosx) dx] = (1/√2)*ln(sec(pi/4 - x) - tan(pi/4 - x))

or

integral [1/(sinx+cosx) dx] = (-1/√2)*ln(sec(pi/4 - x) + tan(pi/4 - x))

(both are the same, you can find out by simplification)


hope this helps.

Answer 2

Hint for the second :
Let tan(x/2)=t
Draw a right angled triangle
Calculate hyp.

Then 1/2* (1+t^2)dx=dt

Hence dx=2dt/(1+t^2)
sinx=2t/(1+t^2)
cosx=(1-t^2)/(1+t^2)
Plug all of these in integral.
Simply and integrate
Remember sinx=2sin(x/2)cos(x/2)
cosx=cos^2(x/2)-sin^2(x/2)

Answer 3

I answer this question assuming that u have min knowledge on integration.
1.u can solve this by using integration by parts
let me explain,
Let u &v b differentiable fns
integral(uv')x dx=(uv)(x) - integral(vu')(x) dx+c
Here we get ,
integral [x tan (x) dx]=x(integral tan x) - tan x(integral x)
=x(log sec x) - tan x +constant
Hence the answer is x(log sec x) -tanx +constant
2.For this I believe u know that sin^2 (x) - cos^2(x)=1
so o can write it as sin^2(x) - cos^2(x)/sinx+cosx
=sin x - cos x
Now find integral of sin x - cos x
= -(cos x+sin x) +constant

Answer 4

2)
introduce sin(pi/4) = cos(pi/4)= 1/sqrt2
(1/sqrt2)integral { 1/ [cos(pi/4)sinx +sin(pi/4)cosx ]} dx

=(1/sqrt2) integral { 1/ sin(x +pi/4) }dx

=(1/sqrt2) integral { cosec(x +pi/4) }dx

=(1/sqrt2) ln{cosec(x +pi/4) - cot(x +pi/4)} + c

other forms of the answer are possible depending on how you substitute cos(pi/4) and sin(pi/4)

general method for integral{1/[asinx+ bcosx]} is given by iyiogrenci (second ans )

if the first qn is integral {x tan(inverse)x} try the following link

http://qwpey.com/mathindex/gl/gl.html

Answer 5

I(xtanx) = x I(tanx) - I(I(tanx)) = xlog(secx) - I(log(secx))