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.