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int(3x+4)cos xdx

∫(3x+4)cos⁡xdx \int(3 x+4) \cos x d x

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Find indefinite integrals using the substitution and by partsright chevron icon

Full solution

Q. ∫(3x+4)cos⁡xdx \int(3 x+4) \cos x d x
  1. Identify integral: Step 11: Identify the integral to solve.\newlineWe need to integrate (3x+4)cos⁡x dx(3x + 4) \cos x \, dx. We'll use integration by parts.\newlineLet u=3x+4u = 3x + 4 and dv=cos⁡x dxdv = \cos x \, dx.\newlineThen, du=3 dxdu = 3 \, dx and v=sin⁡xv = \sin x.
  2. Apply integration by parts: Step 22: Apply the integration by parts formula.\newlineThe formula is ∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du.\newlineSubstituting the values we get:\newline∫(3x+4)cos⁡x dx=(3x+4)sin⁡x−∫sin⁡x⋅3 dx\int(3x + 4) \cos x \, dx = (3x + 4) \sin x - \int\sin x \cdot 3 \, dx
  3. Integrate sin⁡x×3\sin x \times 3: Step 33: Integrate ∫sin⁡x×3 dx\int\sin x \times 3 \, dx.\newlineThis integral is straightforward:\newline∫3sin⁡x dx=−3cos⁡x+C\int 3 \sin x \, dx = -3 \cos x + C
  4. Substitute back: Step 44: Substitute back to the integration by parts formula.\newlinePlugging the integral from Step 33 back into the equation from Step 22:\newline(3x+4)sin⁡x−(−3cos⁡x+C)(3x + 4) \sin x - (-3 \cos x + C)
  5. Simplify expression: Step 55: Simplify the expression.\newlineSimplify the expression to combine like terms:\newline(3x+4)sin⁡x+3cos⁡x+C(3x + 4) \sin x + 3 \cos x + C

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