Questions and detailed answers on Trigonometry integral

Here are sample questions on
trigonometric integrals, along
with detailed answers.

1. Evaluate the integral:
∫(sin^3(x) * cos^2(x)) dx

Solution:
We perform a substitution:
Let u = sin(x), then du = cos(x)
dx.

We rewrite the given integral as:
∫(u^3 * (1 - u^2)) du

Now, we perform polynomial
multiplication:
∫(u^3 - u^5) du

,Integrate the polynomial term-by-
term:
∫u^3 du - ∫u^5 du = (u^4/4) -
(u^6/6) + C
Since u = sin(x), we substitute
back:
(sin^4(x)/4) - (sin^6(x)/6) + C

,So, the final result is:
(sin^4(x)/4) - (sin^6(x)/6) + C

2. Evaluate the integral:
∫(sec^4(x) * tan^2(x)) dx

Solution:
We perform a substitution:
Let u = sec(x), then du =
sec(x)tan(x) dx

We rewrite the given integral as:
∫(u^4 * (u^2 - 1)) u^2 du
Here, we already substituted
sec^2(x) with u^2 - 1.

Now, we perform polynomial

, multiplication:
∫(u^8 - u^6) du

Integrate the polynomial term-by-
term:
∫u^8 du - ∫u^6 du = (u^9/9) -
(u^7/7) + C