Question

Evaluate the integral ∫ sec^2(t)dt / (tan^2(t) + 3tan(t) + 2).

Answer 1

To evaluate the given integral ∫ sec²(t)dt / (tan²(t) + 3tan(t) + 2), we can use a trigonometric substitution. By substituting tan(t) = u, we can simplify the integral and obtain the result tan⁻¹(tan(t)) + C.

Step-by-step explanation:

To evaluate the integral ∫ sec²(t)dt / (tan²(t) + 3tan(t) + 2), we can use a trigonometric substitution. Let's substitute tan(t) = u. Then, dt = du / (1 + u²). Making this substitution, the integral becomes ∫ du / (1 + u²) * sec²(t).

Recall that sec²(t) = 1 + tan²(t). Plugging this into the integral, we have:

∫ du / (1 + u²) * (1 + tan²(t)).

Using the identity sec²(t) = 1 + tan²(t), the integral simplifies to:

∫ du / (1 + u²) * sec²(t) = ∫ du / (1 + u²) = tan⁻¹(u) + C.

Substituting back u = tan(t), the final result is tan⁻¹(tan(t)) + C.