Question

Evaluate the following definite integrals. $$\int_{0}^{2} t e^{t}(\mathbf{i}+2 \mathbf{j}-\mathbf{k}) d t$$

Short answer

Question: Find the value of the definite integral: $$\int_{0}^{2} t e^{t}(\mathbf{i}+2 \mathbf{j}-\mathbf{k}) d t.$$ Answer: The value of the definite integral is: $$(e^2+1)\mathbf{i} + 2e^2\mathbf{j} + (-e^2-1)\mathbf{k}$$

Step-by-step solution

Integrate i component

We need to integrate the i component part of the integral i.e., $$\int_0^2 te^t\mathbf{i} dt$$ Using integration by parts with $$u=t$$ and $$dv=e^t dt$$, the integral can be evaluated as: $$\int te^t\mathbf{i} dt = te^t\mathbf{i} - \int e^t\mathbf{i} dt$$ Now integrating the integral part of the result, we get: $$\int e^t\mathbf{i} dt = e^t\mathbf{i}$$ So the final result of the i component is: $$te^t\mathbf{i} - e^t\mathbf{i}$$, now we need to evaluate it between 0 and 2.

Evaluate i component

By plugging the bounds, we get:: $$(2e^2\mathbf{i} - e^2\mathbf{i}) - (0 - \mathbf{i})$$ $$= e^2\mathbf{i} + \mathbf{i}$$

Integrate j component

Now, let's evaluate the j component, which is: $$\int_0^2 2te^t\mathbf{j} dt$$ Again, using integration by parts with $$u=t$$ and $$dv=2e^t dt$$, we obtain: $$\int 2te^t\mathbf{j} dt = 2te^t\mathbf{j} - \int 2e^t\mathbf{j} dt$$ Now, integrating the integral part of the result, we have: $$\int 2e^t\mathbf{j} dt = 2e^t\mathbf{j}$$ The final result for the j component is: $$2te^t\mathbf{j} - 2e^t\mathbf{j}$$, now we need to evaluate it between 0 and 2.

Evaluate j component

By plugging the bounds, we get: $$(4e^2\mathbf{j} - 2e^2\mathbf{j}) - (0 - 0)$$ $$= 2e^2\mathbf{j}$$

Integrate k component

Finally, let's evaluate the k component: $$\int_0^2 -te^t\mathbf{k} dt$$ Using integration by parts with $$u=-t$$ and $$dv=e^t dt$$, we find: $$\int -te^t\mathbf{k} dt = -te^t\mathbf{k} + \int e^t\mathbf{k} dt$$ Now, integrating the integral part of the result, we obtain: $$\int e^t\mathbf{k} dt = e^t\mathbf{k}$$ The final result for the k component is: $$-te^t\mathbf{k} + e^t\mathbf{k}$$, now we need to evaluate it between 0 and 2.

Evaluate k component

By plugging the bounds, we get: $$(-2e^2\mathbf{k} + e^2\mathbf{k}) - (0 + \mathbf{k})$$ $$= -e^2\mathbf{k} - \mathbf{k}$$

Final Result

Now that we have obtained the values for each component, we can combine the results: $$\int_{0}^{2} t e^{t}(\mathbf{i}+2 \mathbf{j}-\mathbf{k}) d t = (e^2\mathbf{i} + \mathbf{i}) + (2e^2\mathbf{j}) + (-e^2\mathbf{k} - \mathbf{k})$$ The final answer is: $$\boxed{(e^2+1)\mathbf{i} + 2e^2\mathbf{j} + (-e^2-1)\mathbf{k}}$$