Integral Calculus> evaluate the following integral ∫ sin 2x ...

evaluate the following integral∫ sin 2x / a cos^2 (x) + b sin^2 (x) dx

taniska , 10 Years ago

Grade 12

2 Answers

Jitender Singh

Last Activity: 10 Years ago

Ans:
Hello student,
Please find the answer to your question below

$I = \int \frac{sin2x}{acos^{2}x+bsin^{2}x}dx$
$I = \int \frac{sin2x}{a(1-sin^{2}x)+bsin^{2}x}dx$
$I = \int \frac{sin2x}{a+(b-a)sin^{2}x}dx$
$a+(b-a)sin^{2}x = t$
$(b-a)2sinxcosxdx = dt$
$(b-a)sin2x = dt$
$I = \int \frac{dt}{(b-a)t}$
$I = \frac{log(t)}{(b-a)} + c$
$I = \frac{log(a+(b-a)sin^{2}x)}{(b-a)} + c$

Ishu

Last Activity: 6 Years ago

Put a cos²x+bsin²x=t so {a.2cosx(-sinx)+b.2sinxcosx}dx=dt (b-a)sin2xdx=dt i.e. sin 2xdx=dt/b-a int.sin2x/a cos²x+bsin²x dx =int.1/t.dt/b-a =1/b-aint.1/t dt = 1/b-a log| a cos²x+b sin²x|+ c