∫π/3 0 ⟨5sin^4(t)cos(t),−3cos^2(t)sin(t),3tan^2(t)sec^2(t)⟩ dt

To evaluate the vector integral

$\int_{0}^{\frac{\pi}{3}} \langle 5\sin^4(t)\cos(t), -3\cos^2(t)\sin(t), 3\tan^2(t)\sec^2(t) \rangle \, dt,$

we can break it into three separate scalar integrals for each component of the vector.

First component: $\int_0^{\frac{\pi}{3}} 5\sin^4(t)\cos(t) \, dt$

Using the substitution $u = \sin(t)$, which gives $du = \cos(t) \, dt$, we can rewrite the integral as:

$\int_0^{\frac{\pi}{3}} 5\sin^4(t)\cos(t) \, dt = \int_0^{\sin(\frac{\pi}{3})} 5u^4 \, du = \int_0^{\frac{\sqrt{3}}{2}} 5u^4 \, du.$

This evaluates to:

$\frac{5u^5}{5} \Big|_0^{\frac{\sqrt{3}}{2}} = \left(\frac{\sqrt{3}}{2}\right)^5 = \frac{3\sqrt{3}}{32}.$

Second component: $\int_0^{\frac{\pi}{3}} -3\cos^2(t)\sin(t) \, dt$

Using the substitution $u = \cos(t)$, with $du = -\sin(t) \, dt$, we get:

$\int_0^{\frac{\pi}{3}} -3\cos^2(t)\sin(t) \, dt = \int_1^{\frac{1}{2}} 3u^2 \, du = \left[ u^3 \right]_1^{\frac{1}{2}} = \frac{1}{8} - 1 = -\frac{7}{8}.$

Third component: $\int_0^{\frac{\pi}{3}} 3\tan^2(t)\sec^2(t) \, dt$

Recognize that $\int \tan^2(t)\sec^2(t) \, dt$ is just a standard integral:

$\int \tan^2(t)\sec^2(t) \, dt = \frac{\tan^3(t)}{3}.$

Evaluating this from $0$ to $\frac{\pi}{3}$:

$\left[ \tan^3(t) \right]_0^{\frac{\pi}{3}} = \tan^3\left(\frac{\pi}{3}\right) = \left(\sqrt{3}\right)^3 = 3\sqrt{3}.$

Final Result

Thus, the evaluated integral is:

$\int_{0}^{\frac{\pi}{3}} \langle 5\sin^4(t)\cos(t), -3\cos^2(t)\sin(t), 3\tan^2(t)\sec^2(t) \rangle \, dt = \langle \frac{3\sqrt{3}}{32}, -\frac{7}{8}, 3\sqrt{3} \rangle.$

Would you like further explanation or details?

Here are some related questions to expand your understanding:

1. How do you solve integrals involving trigonometric identities?
2. What is the role of substitution in evaluating trigonometric integrals?
3. Can you explain how trigonometric functions behave over different intervals?
4. How can we simplify powers of trigonometric functions before integration?
5. What are common techniques for solving vector integrals?

Tip: Always check if substitution simplifies the integral before using trigonometric identities!