[tex]\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=\int_{t=0}^{t=1}\mathbf f(\mathbf r(t))\cdot\mathrm d(t^2\,\mathbf i+t^3\,\mathbf j+t^2\,\mathbf k)[/tex]
[tex]=\displaystyle\int_0^1\bigg((t^2+t^3)\,\mathbf i+(t^3-t^2)\,\mathbf j+t^4\,\mathbf k\bigg)\cdot\bigg(2t\,\mathbf i+3t^2\,\mathbf k+2t\,\mathbf k\bigg)\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1(5t^5-t^4+2t^3)\,\mathrm dt=\dfrac{17}{15}[/tex]
