Evaluate the integral: $\int -5 \sin^5(x) \, dx$

Note: Use an upper-case "C" for the constant of integration.

Question :

Evaluate the integral: $\int -5 \sin^5(x) \, dx$

note: use an upper-case "c" for the constant of integration.

![Evaluate the integral: $\int -5 \sin^5(x) \, dx$

note: use an upper-case " | Doubtlet.com](https://doubt.doubtlet.com/images/20241111-133234-3.19.png)

Solution:

Neetesh Kumar | November 11, 2024

Calculus Homework Help

This is the solution to DHW Calculus
Assignment: 3 Question Number 19
Contact me if you need help with Homework, Assignments, Tutoring Sessions, or Exams for STEM subjects.
You can see our Testimonials or Vouches from here of the previous works I have done.

Get Homework Help

Step-by-step solution:

To evaluate this integral, we need to apply the reduction formula for powers of sine functions.

Step 1: Apply the power reduction identity

We can express $\sin^5(x)$ in terms of lower powers of sine using the standard reduction formula for odd powers of sine:

$\sin^5(x) = \sin(x) \cdot \sin^4(x)$

Now, using the reduction identity:

$\sin^4(x) = \left(1 - \cos^2(x)\right)^2$

Thus, we rewrite the integral as:

$\int -5 \sin^5(x) \, dx = -5 \int \sin(x) (1 - \cos^2(x))^2 \, dx$

Step 2: Perform substitution

Let $u = \cos(x)$ . Then $du = -\sin(x) \, dx$ . Substituting these into the integral:

$-5 \int \sin(x) (1 - \cos^2(x))^2 \, dx = 5 \int (1 - u^2)^2 \, du$

Step 3: Expand and simplify the integrand

Expanding $(1 - u^2)^2$ :

$(1 - u^2)^2 = 1 - 2u^2 + u^4$

Now, the integral becomes:

$5 \int (1 - 2u^2 + u^4) \, du$

Step 4: Integrate term by term

Now, we can integrate each term:

$5 \int (1 - 2u^2 + u^4) \, du = 5 \left( u - \frac{2u^3}{3} + \frac{u^5}{5} \right) + C$

Step 5: Substitute back $u = \cos(x)$

Substituting back $u = \cos(x)$ , we get:

$5 \left( \cos(x) - \frac{2\cos^3(x)}{3} + \frac{\cos^5(x)}{5} \right) + C$

Thus, the final answer is: