what do you get when you:

integrate 10sin^4xcosx dx

when you integrate 9x^2e^6x^3 dx

To integrate the function 10sin^4(x)cos(x) dx, you can use the method of trigonometric identities to simplify the expression. Recall the double angle formula:

sin(2θ) = 2sin(θ)cos(θ)

We can rewrite sin^4(x) as (sin^2(x))^2 and apply the double angle formula twice to convert it to a product of trigonometric functions involving cos(x) and sin(x).

sin^4(x) = (sin^2(x))^2 = (1 - cos^2(x))^2 = (1 - 2cos^2(x) + cos^4(x))

Now, substitute this expression in the original integral:

∫ 10sin^4(x)cos(x) dx = ∫ 10(1 - 2cos^2(x) + cos^4(x))cos(x) dx

Expanding the expression, we get:

∫ (10cos(x) - 20cos^3(x) + 10cos^5(x)) dx

Splitting this integral into separate terms, you can integrate each term separately:

∫ 10cos(x) dx - ∫ 20cos^3(x) dx + ∫ 10cos^5(x) dx

The integration of cos(x) is straightforward, giving sin(x):

10∫ cos(x) dx = 10sin(x) + C1

To integrate cos^3(x), you can use the reduction formula:

∫ cos^n(x) dx = (1/n)cos^(n-1)(x)sin(x) + ((n-1)/n)∫ cos^(n-2)(x) dx

Applying the reduction formula twice for cos^3(x), it becomes:

∫ 20cos^3(x) dx = 20/3*cos^2(x)sin(x) + (2/3)∫ cos(x) dx

Using the previous result, we know that:

∫ cos(x) dx = sin(x) + C2

Substituting this, we get:

20/3*cos^2(x)sin(x) + (2/3)∫ cos(x) dx = 20/3*cos^2(x)sin(x) + (2/3)(sin(x) + C2)

Lastly, to integrate cos^5(x), we can again use the reduction formula:

∫ cos^5(x) dx = (4/5)cos^4(x)sin(x) + (2/5)∫ cos^2(x) dx

Since we already have a formula for cos^2(x), we can substitute it:

∫ cos^5(x) dx = (4/5)cos^4(x)sin(x) + (2/5)(1/2)(x + sin(x)) + C3

Now, substitute the results back into the original integral:

∫ 10sin^4(x)cos(x) dx = 10sin(x) - 20/3*cos^2(x)sin(x) - (2/3)sin(x) + (4/5)cos^4(x)sin(x) + (2/5)(1/2)(x + sin(x)) + C

Simplifying this expression gives the final result.