Evaluate the indefinite integral of -8*cos(0.8*t^2) with respect to t.

Step-by-step solution

1. 1

   Recognise that no elementary antiderivative exists. By Liouville's theorem, $\cos(at^{2})$ has no antiderivative expressible with elementary functions — the same obstruction that makes $\int e^{-t^{2}}dt$ non-elementary. So the goal is not to "find" a formula by substitution but to express the answer through the standard special function built for it.

2. 2

   Recall the definition of the Fresnel cosine integral. The standard normalisation is

   $C(x)=\int_{0}^{x}\cos\!\left(\frac{\pi u^{2}}{2}\right)du,$

   with $C(0)=0$. Every integral of the form $\int\cos(at^{2})\,dt$ can be rescaled into this shape; the only work is finding the right change of variable.

3. 3

   Rescale the argument. Write $0.8=\tfrac45$ and look for $u=\lambda t$ with $\frac{\pi u^{2}}{2}=\frac45 t^{2}$. That forces

   $\frac{\pi\lambda^{2}}{2}=\frac{4}{5}\;\Longrightarrow\;\lambda=\sqrt{\frac{8}{5\pi}},$

   and since $dt=\frac{du}{\lambda}$, in general

   $\int\cos(at^{2})\,dt=\sqrt{\frac{\pi}{2a}}\;C\!\left(\sqrt{\frac{2a}{\pi}}\,t\right)+C_{1}.$

4. 4

   Substitute $a=\tfrac45$ and simplify the constant. The prefactor is

   $\sqrt{\frac{\pi}{2\cdot\frac45}}=\sqrt{\frac{5\pi}{8}},$

   so multiplying by the $-8$ in front of the integrand:

   $-8\sqrt{\frac{5\pi}{8}}=-\frac{8}{2\sqrt2}\sqrt{5\pi}=-2\sqrt{2}\sqrt{5\pi}=-2\sqrt{10\pi}.$

   That simplification, $8/\sqrt8=\sqrt8=2\sqrt2$, is the only algebra that can go wrong here.

5. 5

   Write the answer and sanity-check the derivative. The result is

   $\int-8\cos(0.8t^{2})\,dt=-2\sqrt{10\pi}\;C\!\left(\sqrt{\frac{8}{5\pi}}\,t\right)+C_{1}.$

   Differentiating: $\frac{d}{dt}C(\lambda t)=\lambda\cos\!\left(\frac{\pi\lambda^{2}t^{2}}{2}\right)=\sqrt{\tfrac{8}{5\pi}}\cos(0.8t^{2})$, and $-2\sqrt{10\pi}\cdot\sqrt{\tfrac{8}{5\pi}}=-2\sqrt{16}=-8$ ✓. If the comma in "0,8" was meant as a thousands separator the integrand would instead be $\cos(8t^{2})$, giving $-2\sqrt{\pi}\,C\!\left(\tfrac{4t}{\sqrt{\pi}}\right)$.