[tex]\mathbf r(t)=x(t)\,\mathbf i+y(t)\,\mathbf j+z(t)\,\mathbf k[/tex]
[tex]\mathrm d\mathbf r=(x'(t)\,\mathbf i+y'(t)\,\mathbf j+z'(t)\,\mathbf k)\,\mathrm dt[/tex]
[tex]\implies\mathrm d\mathbf r=(5t^4\,\mathbf i-4t^3\,\mathbf j+3t^2\,\mathbf k)\,\mathrm dt[/tex]
[tex]\mathbf f(\mathbf r(t))=3\sin t^5\,\mathbf i-4\cos(-t^4)\,\mathbf j-t^8\,\mathbf k[/tex]
[tex]\mathbf f\cdot\mathrm d\mathbf r=15t^4\sin t^5+16t^3\cos t^4-3t^{10}[/tex]
[tex]\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=\int_{t=0}^{t=1}(15t^4\sin t^5+16t^3\cos t^4-3t^{10})\,\mathrm dt[/tex]
[tex]=-3\cos t^5+4\sin t^4-\dfrac3{11}t^{11}\bigg|_{t=0}^{t=1}[/tex]
[tex]=\left(-3\cos1+4\sin1-\dfrac3{11}\right)-\left(-3+0-0\right)[/tex]
[tex]=4\sin1-3\cos1+\dfrac{30}{11}[/tex]
