Sir, I have been doing a proof related to one research topic. But after a long effort, I got ended up in a messy integration equation. Could you give me some suggestions to solve this equations? (Any method like substitutions etc are welcome). Kind of stucked my work because of this. I am giving you the integration equation as follows
NB :: "*" notation also implies multiplication. Means $a \times b = ab=a*b$
Given Data in the question
$ \lambda(t) = \left(\sqrt{\left( \frac{t}{2}A- a \right )^2+\left( \frac{t }{2}B- b \right )^2+\left( \frac{t }{2}C- c \right )^2} \right) \\= \sqrt{ \frac{t^2}{4}(A^2+B^2+C^2)-(Aa+Bb+Cc)t+(a^2+b^2+c^2) } \tag 1$
We here imply only positive square root. A,B,C,a,b and c are constants we cant alter the values. Only t is a variable here.
2. $\phi(t)_0= \frac{sin \left( \frac{t}{2}\lambda(t) \right)}{ \lambda(t)}*\left( \frac{t }{2}A- a \right) \tag2$ $\phi(t)_1= \frac{sin \left( \frac{t}{2}\lambda(t) \right)}{ \lambda(t)}*\left( \frac{t }{2}B- b \right) \tag3$ $\phi(t)_2= \frac{sin \left( \frac{t}{2}\lambda(t) \right)}{ \lambda(t)}*\left( \frac{t }{2}C- c \right) \tag2$
We have two sets of questions. SET 1 is the original problem. I have added SET 2 as supplementary because solving this will lead to the solution of SET 1. Any solution to either set 1 or set 2 is welcome. Main issue I face here is the difficulty to take out square root from sin . Note that A,B,C,a,b,c can have any real values. It is not necessary that the $\frac{t^2}{4}(A^2+B^2+C^2)-(Aa+Bb+Cc)t+(a^2+b^2+c^2) $ has equal roots
Question
SET 1
- Is there anyway to solve $ \int \phi(t)_0 cos \left( \frac{t}{2}\lambda(t)\right) \ dt\tag 5$?
- Is there anyway to solve $ \int \phi(t)_0 \phi(t)_1 \ dt \tag 6$ ?
- Is there anyway to solve $ \int \phi(t)_1 \phi(t)_2 \ dt \tag 7$ ?
SET 2
- Is there anyway to solve $\displaystyle \int t^2\frac{\sin^2\left(t \sqrt{ at^2+bt+c}\right) }{ at^2+bt+c} \operatorname{d}t \tag8$?
Is there anyway to solve $\displaystyle \int t\frac{\sin^2\left(t \sqrt{ at^2+bt+c}\right) }{ at^2+bt+c } \operatorname{d}t \tag8$?
Is there anyway to solve $\displaystyle \int t^2\frac{\sin \left(2\times t\sqrt{ at^2+bt+c}\right) }{ at^2+bt+c } \operatorname dt \tag8$?
Is there anyway to solve $\displaystyle \int t \frac{\sin \left(2\times t\sqrt{ at^2+bt+c}\right) }{ \sqrt{ at^2+bt+c}} \operatorname dt \tag8$?
Some attempts currently I am trying
Thinking about the possiblity of exponent expression of sin x and cos x as here
Thinking about the possibility of product form results of sin xcos x as here
Thinking about substitution method by utilizing property $ax^2+bx+c =a \left(x+ \frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}$ as done here
NB : So far I couldnot solve it. Main issue here is the square root inside trigonometric terms. It is difficult to convert it in to a solvable propblem
Thanks for taking time to read my doubt. Hope nice suggestions and discussions