6x sin(5x) - cos(5x) + C
Added by Robert V.
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d/dx (6x sin(5x)) = 6(sin(5x) + 5x cos(5x)) Show more…
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Evaluate the integral: ∫ 20x^2 sin(πx) dx Step 1: To use the integration-by-parts formula ∫ u dv = uv - ∫ v du, we must choose one part of ∫ 20x^2 sin(πx) dx to be u, with the rest becoming dv. Since the goal is to produce a simpler integral, we will choose u = 20x^2. This means that dv = sin(πx) dx. Step 2: Now, since u = 20x^2, then du = 40x dx. Step 3: With our choice that dv = sin(πx) dx, then v = ∫ sin(πx) dx, which can be calculated using the substitution w = πx. In this case, we have dx = (1/π) dw, and we get v = -cos(πx)/π (in terms of x). Step 4: Now, the integration-by-parts formula ∫ u dv = uv - ∫ v du gives us ∫ u dv = uv - ∫ v du = -20/π x^2 cos(πx) + 40/π ∫ x cos(πx) dx. We must repeat integration-by-parts to do this second integral. Using the variables U and V, we choose U = x, which will give dU = dx, and dV = cos(πx) dx, which will give V = ∫ cos(πx) dx = sin(πx)/π.
Sri K.
Evaluate the integral. ∫ 3x cos(4x) dx To use the integration-by-parts formula ∫ u dv = uv - ∫ v du, we must choose one part of ∫ 3x cos(4x) dx to be u, with the rest becoming dv. Since the goal is to produce a simpler integral, we will choose u = 3x. This means that dv = cos(4x) dx. Now, since u = 3x, then du = 3 dx. With our choice that dv = cos(4x) dx, then v = ∫ cos(4x) dx. This can be calculated using integration by substitution. In this case (ignoring the constant of integration) we get v = 1/4 sin(4x). Now, the integration-by-parts formula ∫ u dv = uv - ∫ v du gives us ∫ 3x cos(4x) dx = 3x (1/4 sin(4x)) - ∫ 1/4 sin(4x) (3 dx). We must use substitution to do this second integral. We can use the substitution t = 4x, which will give dx = 1/4 dt. Ignoring the constant of integration, we have ∫ sin(4x) dx = -1/4 cos(4x). Combining our results and including the constant of integration, C, we finally get ∫ 3x cos(4x) dx = 3/4 x sin(4x) + 3/16 cos(4x) + C.
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