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$\begin{align} (1)\quad &\int\dfrac{1}{\sqrt{x^4+1}}\mathrm dx\\ (2)\quad &\int\dfrac{\sin x}{x}\mathrm dx \end{align}$

How can I solve these types of integrals?

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google.com/… –  JavaMan Aug 1 '11 at 18:40
What have you tried? Is this homework? –  JavaMan Aug 1 '11 at 18:42
@DJC I doubt that this is homework. Would take an evil teacher to assign these "problems" :-) (But of course, point taken: the OP should've put more effort in framing the question.) –  Srivatsan Aug 1 '11 at 19:02
"Solve" is the wrong word here. "Evaluate" is appropriate. –  Michael Hardy Aug 1 '11 at 19:18
@Srivatsan: I agree, and that's a fair point, though I suppose that might depend on the level of the class. Nonetheless, the question is not well motivated. –  JavaMan Aug 1 '11 at 19:21

2 Answers 2

These integrals are not "elementary". The first one is basically an elliptic integral of the first kind, which can be explicitly written in terms of hypergeometric functions after some manipulations (see Euler's integral representation of hypergeometric functions).

The second one is the sine integral, which also has no representation in terms of elementary functions.

Hope this helps!

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Bruno already mentioned that both of the OP's integrands are not expressible in terms of "usual" functions; I'll address the "how can I evaluate" portion of the question for the first integral.

Letting $x=\tan\frac{u}{2}$, $\mathrm dx=\frac{\mathrm du}{1+\cos\,u}$ (the Weierstrass substitution), we get

$$\int \frac1{(1+\cos\,u)\sqrt{1+\tan^4\frac{u}{2}}}\mathrm du$$

$$\int \frac1{\sqrt{(1+\cos\,u)^2\left(1+\left(\frac{1-\cos\,u}{1+\cos\,u}\right)^2\right)}}\mathrm du$$

which simplifies to

$$\int\frac{\mathrm du}{\sqrt{4-2\,\sin^2 u}}$$

Removing a factor of $\frac12$ from the integrand gives

$$\frac12\int\frac{\mathrm du}{\sqrt{1-\frac12\sin^2 u}}$$

which is easily recognized as the incomplete elliptic integral of the first kind:

$$\frac12 F\left(u\mid\frac12\right)$$

Undoing the initial Weierstrass substitution yield the final result,

$$\frac12 F\left(2\arctan\,x\mid\frac12\right)$$

Alternatively, one could have instead used $x=\cot\frac{u}{2}$ as the substitution, but I'll leave that as an exercise...

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