Integrate the following indefinite integral:
(sin7x)^12 * (cos7x)^3
Hint: sin^2 + cos^2 = 1
= int [(sin7x)^12 * (cos7x)^2 * cos7x *dx]
= int [(sin7x)^12 * (1-sin 7x)^2 * cos7x *dx]
=int [(sin7x)^12 * cos 7x * dx] - int [(sin7x)^12 * (sin 7x)^2 * cos7x *dx]
=int[u^12*(1/7)*du] - int[u^14*(1/7)*du]
=(1/7)*(1/13)*u^13 - (1/7)*(1/15)*u^15 + C
=(1/7)*u^13*( (1/13) - (1/15)*u^2 ) + C
=(1/7)*(sin 7x)^13 * ( (1/13) - (1/15)*(sin7x)^2 ) + C
THANK YOU SOOOOOOOOOOOOOO MUCH!~!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
To integrate the given expression, we can make use of the trigonometric identity: sin^2(x) + cos^2(x) = 1. Since we have sin^12(x) and cos^3(x), we can rewrite the expression in terms of just sin(x) or cos(x) using this identity.
Let's start by rewriting the expression in terms of sin(x):
(sin^2(x))^6 * cos^3(x) = (1 - cos^2(x))^6 * cos^3(x)
Now, let's substitute u = cos(x). This will simplify the integral as we are left with a single variable:
du = -sin(x) dx
Now, we need to replace the remaining sin(x) term in the expression with u:
(1 - u^2)^6 * u^3
The indefinite integral of this expression is now easier to compute. We can expand the polynomial term and then integrate each term separately.
Using the Binomial Theorem, we can expand (1 - u^2)^6:
(1 - u^2)^6 = 1 - 6u^2 + 15u^4 - 20u^6 + 15u^8 - 6u^10 + u^12
Now, we can integrate each term using the power rule:
∫ (1 - u^2)^6 * u^3 du = ∫ (u^3 - 6u^5 + 15u^7 - 20u^9 + 15u^11 - 6u^13 + u^15) du
Using the power rule, the integral of each term becomes:
u^4/4 - 6u^6/6 + 15u^8/8 - 20u^10/10 + 15u^12/12 - 6u^14/14 + u^16/16 + C
Simplifying each term, we get:
u^4/4 - u^6 + 5u^8/8 - 2u^10/5 + 5u^12/12 - 3u^14/7 + u^16/16 + C
Now, substitute back u = cos(x) to get the final result:
∫ (sin^7(x))^12 * (cos^7(x))^3 dx = ∫ (cos(x))^4/4 - (cos(x))^6 + 5(cos(x))^8/8 - 2(cos(x))^10/5 + 5(cos(x))^12/12 - 3(cos(x))^14/7 + (cos(x))^16/16 + C
Therefore, the result of the indefinite integral is:
(cos(x))^4/4 - (cos(x))^6 + 5(cos(x))^8/8 - 2(cos(x))^10/5 + 5(cos(x))^12/12 - 3(cos(x))^14/7 + (cos(x))^16/16 + C, where C is the constant of integration.