# calculus

posted by .

∫ sin(x)sin(3x)dx

find the integral

• calculus -

one way:

sin(3x) = sinx cos2x + cosx sin2x
= sinx(2cos^2(x)-1) + 2cosx sinx cosx
= cos^2(x) sinx - sinx + 2cos^2(x) sinx
= 3cos^2(x)sinx - sinx

∫ = -cos^3(x) + cos(x)

another way:

sinx sin3x = 1/2 cos(2x) - 1/2 cos(4x)
∫ = 1/4 sin2x - 1/8 sin4x

a little manipulation of that also yields

sin^3(x) - cos(x)

the various expressions are not identical, but differ only by a constant C

## Similar Questions

1. ### Integral

That's the same as the integral of sin^2 x dx. Use integration by parts. Let sin x = u and sin x dx = dv v = -cos x du = cos x dx The integral is u v - integral of v du = -sinx cosx + integral of cos^2 dx which can be rewritten integral …
2. ### Math integrals

What is the indefinite integral of ∫ [sin (π/x)]/ x^2] dx ?
3. ### calc

find integral using table of integrals ) integral sin^4xdx this the formula i used integral sin^n xdx =-1/n sin^n-1xcosx +n-1/n integral sin^n-2 using the formula this is what i got: integral sin^4xdx=-1/4sin^3xcosx+3/4 integral sin^2xdx= …
4. ### Calculus AP

Evaluate the integral interval from [0 to pi] t sin(3t)dt Use integration by parts u=t and dv=sin(3t)dt. then du=dt and v=-cos(3t)/3 here is my problem but Im having problem to solve with pi. ∫t sin(3t)dt = -tcos(3t)/3 - ∫[-cos(3t)/3]dt …
5. ### Definite integral by parts (correction)

Hello, I just wanted to verify if my work was good. Calculate the following integral by parts: ∫ upper limit is 1/5 and lower limit is 1/10. of 10sin^-1 (5x)dx so first I named the variables: u = 10 sin^-1 (5x) du = 50 / sqr(1-25x^2) …
6. ### Calculus

Hello, I just wanted to verify if my work was good. Calculate the following integral by parts: ∫ upper limit is 1/5 and lower limit is 1/10. of 10sin^-1 (5x)dx so first I named the variables: u = 10 sin^-1 (5x) du = 50 / sqr(1-25x^2) …
7. ### Calculus

Alright, I want to see if I understand the language of these two problems and their solutions. It asks: If F(x) = [given integrand], find the derivative F'(x). So is F(x) just our function, and F'(x) our antiderivative?
8. ### Calculus

Evaluate ∫ (cos(x))^(1/2)sin(x)dx Let u = cos(x)?
9. ### Calculus

∫((cos^3(x)/(1-sin^(2)) What is the derivative of that integral?
10. ### Calculus

Integrate 1/sinx dx using the identity sinx=2(sin(x/2)cos(x/2)). I rewrote the integral to 1/2 ∫ 1/(sin(x/2)cos(x/2))dx, but I don't know how to continue. Thanks for the help. Calculus - Steve, Tuesday, January 12, 2016 at 12:45am …

More Similar Questions