I am bit stucked with an integration form while doing one of my proofs for a graphics application.Issue is I cant take out the terms from the trigonometric functions for a proper known integral format. Could you give me some suggestions

Problem 1

$\displaystyle\int\frac{\sin^2\left( \sqrt{ ax^2+bx+c}\right) }{ \sqrt{ ax^2+bx+c}} \operatorname{d}x \tag1$

Problem 2

$\displaystyle\int\frac{\sin \left(2\times \sqrt{ ax^2+bx+c}\right) }{ \sqrt{ ax^2+bx+c}} \operatorname dx \tag2$

  • $\begingroup$ A search on Wolfram Alpha wasn't able to find a closed form solution. $\endgroup$ – user28375028 Sep 15 '14 at 2:55
  • $\begingroup$ It has no direct solution. I feel we need to do some manipulation before applying knowing known integrals $\endgroup$ – Nirvana Sep 15 '14 at 2:57


For $\int\dfrac{\sin^2\sqrt{ax^2+bx+c}}{\sqrt{ax^2+bx+c}}dx$ ,





For $\int\dfrac{\sin\left(2\sqrt{ax^2+bx+c}\right)}{\sqrt{ax^2+bx+c}}dx$ ,





I do not think taht any closed form could be found for these integrals except in the case where $b^2-4ac=0$ where we should find $$a x^2+bx+x=a \Big(x+\frac{b}{2a}\Big)^2$$ $$\sqrt{a x^2+bx+x}=\sqrt a (x+\frac{b}{2a})$$ For such a case $$\displaystyle\int\frac{\sin^2\left( \sqrt{ ax^2+bx+c}\right) }{ \sqrt{ ax^2+bx+c}} dx =\frac{\log \left(\frac{2 a x+b}{2 \sqrt{a}}\right)-\text{Ci}\left(\frac{b+2 a x}{\sqrt{a}}\right)}{2 \sqrt{a}}$$ $$\displaystyle\int\frac{\sin \left(2\times \sqrt{ ax^2+bx+c}\right) }{ \sqrt{ ax^2+bx+c}} dx=\frac{\text{Si}\left(\frac{b+2 a x}{\sqrt{a}}\right)}{\sqrt{a}}$$


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