# How do you find the integral of #x(sin^2(ax))#?

This equivalency is used to transform the integral into:

Dividing the integral:

The power rule for integration makes it simple to complete the first integral:

Using the following in the formula:

Should we seek a shared denominator:

By signing up, you agree to our Terms of Service and Privacy Policy

To find the integral of x(sin^2(ax)), we can use integration by parts. Let u = x and dv = sin^2(ax) dx. Then, du = dx and v = (1/2)(x - (1/(2a))sin(2ax)). Applying the integration by parts formula, we have:

∫ x(sin^2(ax)) dx = (1/2)x^2 - (1/(2a)) ∫ x sin(2ax) dx

To evaluate the remaining integral, we use integration by parts again. Let u = x and dv = sin(2ax) dx. Then, du = dx and v = -(1/(2a))cos(2ax). Applying the integration by parts formula, we have:

∫ x sin(2ax) dx = -(1/(4a))x cos(2ax) + (1/(4a^2)) ∫ cos(2ax) dx

= -(1/(4a))x cos(2ax) + (1/(8a^3))sin(2ax) + C

Substituting this result back into the original equation, we have:

∫ x(sin^2(ax)) dx = (1/2)x^2 - (1/(2a))(-(1/(4a))x cos(2ax) + (1/(8a^3))sin(2ax)) + C

= (1/2)x^2 + (1/(8a^2))sin(2ax) - (1/(8a^3))x cos(2ax) + C

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7