Integral sinx raise power n dx
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Step-by-step solution
8 stepsThis is a request for the integral of sin(x) raised to the power of n, with respect to x. This is typically solved using integration by parts and results in a reduction formula.
Define the integral.
Let Iₙ = ∫ sinⁿ(x) dx
Use integration by parts.
We can rewrite sinⁿ(x) as sinⁿ⁻¹(x) · sin(x).
Let u = sinⁿ⁻¹(x) and dv = sin(x) dx.
Then du = (n-1)sinⁿ⁻²(x)cos(x) dx and v = -cos(x).Apply the integration by parts formula
∫ u dv = uv - ∫ v du.
Iₙ = sinⁿ⁻¹(x) · (-cos(x)) - ∫ (-cos(x)) · (n-1)sinⁿ⁻²(x)cos(x) dx
Iₙ = -sinⁿ⁻¹(x)cos(x) + (n-1) ∫ cos²(x)sinⁿ⁻²(x) dxUse the identity cos²(x) = 1 - sin²(x).
Iₙ = -sinⁿ⁻¹(x)cos(x) + (n-1) ∫ (1 - sin²(x))sinⁿ⁻²(x) dx
Iₙ = -sinⁿ⁻¹(x)cos(x) + (n-1) ∫ (sinⁿ⁻²(x) - sinⁿ(x)) dxSplit the integral.
Iₙ = -sinⁿ⁻¹(x)cos(x) + (n-1) ∫ sinⁿ⁻²(x) dx - (n-1) ∫ sinⁿ(x) dx
Rearrange the equation to solve for Iₙ.
Notice that the last term is -(n-1)Iₙ.
Iₙ = -sinⁿ⁻¹(x)cos(x) + (n-1)Iₙ₋₂ - (n-1)IₙAdd (n-1)Iₙ to both sides.
Iₙ + (n-1)Iₙ = -sinⁿ⁻¹(x)cos(x) + (n-1)Iₙ₋₂
nIₙ = -sinⁿ⁻¹(x)cos(x) + (n-1)Iₙ₋₂Divide by n to get the reduction formula.
Iₙ = (-1/n)sinⁿ⁻¹(x)cos(x) + ((n-1)/n)Iₙ₋₂
The reduction formula for the integral of sinⁿ(x) dx is:
∫ sinⁿ(x) dx = (-1/n)sinⁿ⁻¹(x)cos(x) + ((n-1)/n) ∫ sinⁿ⁻²(x) dx
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