Integration of x^(1/2) sinx

Question

My book say that integration of $x^{1/2} \sin x$ is not possible, why is it so? Which functions do not have an anti derivative?
Does it mean that they do not have any area under the curve?
But that's not true since the graph says different.

(Source: https://www.desmos.com/calculator)

When is a function not integrable?

Answer 1

$$\int x^{1/2}\sin(x)dx=\int x^{1/2}\sum_{k=0}^\infty\frac{(-1)^kx^{2k+1}}{(2k+1)!} dx=\sum_{k=0}^\infty\frac{(-1)^kx^{2k+5/2}}{(2k+\frac52)(2k+1)!}.$$

The series converges for all $x$.

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By substitution then by parts, you have

$$\int x^{1/2}\sin(x)dx=2\int t^2\sin(t^2)dt=-t\cos(t^2)+\int \cos(t^2)dt,$$ where the last integral is known as the Fresnel cosine function.

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To get a numerical value of the definite integral, Simpson's method will perform better.

Answer 2

There is an important distinction between integrable and expressible in terms of elementary functions. Integrable means that the integral exists. If we take one of the endpoints to be variable then we get the anti derivative. All continuous functions such as the one you mention are integrable. The question of whether this antiderivative can be written as a combination of the usual rational, algebraic, exponential, logarithmic, and trigonometric functions is different. The antiderivitive of your function exists, but cannot be so expressed.