[tex]\mathbf f(x,y,z)=3\sin x\,\mathbf i-\cos y\,\mathbf j-5xz\,\mathbf k[/tex]
[tex]\mathbf r(t)=\mathbf r'(t)\,\mathrm dt=t^5\,\mathbf i-t^4\,\mathbf j+t^3\,\mathbf k[/tex]
[tex]\implies\mathrm d\mathbf r(t)=(5t^4\,\mathbf i-4t^3\,\mathbf j+3t^2\,\mathbf k)\,\mathrm dt[/tex]
[tex]\displaystyle\int_C\mathbf f(x,y,z)\cdot\mathrm d\mathbf r=\int_{t=0}^{t=1}f(x(t),y(t),z(t))\cdot\mathbf r'(t)\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1(3\sin(t^5)\,\mathbf i-\cos(t^4)\,\mathbf j-5t^8\,\mathbf k)\cdot(5t^4\,\mathbf i-4t^3\,\mathbf j+3t^2\,\mathbf k)\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1(15t^4\sin(t^5)+4t^3\cos(t^4)-15t^{10})\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1\bigg(3\sin(t^5)\,\mathrm d(t^5)+\cos(t^4)\,\mathrm d(t^4)-15t^{10}\,\mathrm dt\bigg)[/tex]
[tex]=-3\cos(t^5)+\sin(t^4)-\dfrac{15}{11}t^{11}\bigg|_{t=0}^{t=1}[/tex]
[tex]=-3\cos1+\sin1+\dfrac{18}{11}[/tex]
