Students Adam, Bob, Chris, and David were tasked with solving the indefinite integral: $$\int \sin(x^2 )\mathrm{d}x$$
Adam considered the integral simple and solved it immediately: $$\int \sin (x^2 )\mathrm{d}x =-\cos (x^2 )+c$$
Bob realized that it was an integral of a composite function and solved it as follows: $$\int \sin (x^2 )\mathrm{d}x =-\cos (x^2 ) \cdot 2x+c $$
Chris recognized that it was not a straightforward integral and proceeded in the following way: $$ \begin{gather}\int \sin(x^2 )\mathrm{d}x =\int [1-\cos(x^2 ) ]\mathrm{d}x =x-\int \cos(x^2 )\mathrm{d}x = \cr =x-\int [1-\sin(x^2 ) ]\mathrm{d}x =x-x-\int \sin(x^2 )\mathrm{d}x \end{gather} $$ He ended up with the same integral on both sides and rearranged the equation into the form: $$ 2\int \sin(x^2 )\mathrm{d}x =x-x=0, $$ concluding that the integral was zero.
David decided to use the substitution $x^2=t$ and found the relation between differentials $2x\mathrm{d}x =\mathrm{d}t$, from which he derived $\mathrm{d}x =\frac{\mathrm{d}t}{2x}$. Then, he proceeded as follows: $$\int \sin(x^2 )\mathrm{d}x =\int \sin t \frac{\mathrm{d}t}{2x} $$ He realized that $\frac{1}{2x}$ could be factored out because he integrated with respect to the variable t and continued: $$\int \sin t \frac{\mathrm{d}t}{2x} = \frac{1}{2x} \int \sin t\, \mathrm{d}t= \frac{1}{2x} (-\cos t ) $$ Finally, he returned to the original variable ($t=x^2$ ), added the integration constant, and got the resulting antiderivative: $$-\frac{1}{2x} \cos(x^2)+c $$ Did anyone solve the integral correctly?
Nobody
Adam
Bob
Chris
David
While the indefinite integral $$\int \sin(x^2 )\mathrm{d}x $$ might seem simple at first glance, it cannot be solved using standard methods. Specifically, it is not possible to express in terms of elementary functions. The sought antiderivative is the Fresnel integral $S(x)$ defined by the formula: $$ S(x)=∫_0^x \sin(t^2 )\mathrm{d}t $$ In other words, it is the function defined precisely by the integral we were tasked to solve. Each student made a fundamental mistake in their solution. For example, Adam did not consider the argument $x^2$ of the function $\sin x^2$ at all. Bob incorrectly combined the integration with the differentiation of the inner function and Chris did not understand the difference between $\sin^2 x$ and $\sin(x^2)$ which led to an incorrect interpretation of the trigonometric identity. Chris also made a mistake in the sign before the integral. David factored out of the integral an expression containing the variable, which is not permissible.