9–40. Integration by parts Evaluate the following integrals using integration by parts.
29. ∫ e⁻ˣ sin(4x) dx

Guida verificata passo dopo passo

Identify the integral to solve: $\int e^{-x} \sin(4x) \, dx$.

Recall the integration by parts formula: $\int u \, dv = uv - \int v \, du$.

Choose $u$ and $dv$ wisely. For this integral, let $u = \sin(4x)$ and $dv = e^{-x} dx$.

Compute $du$ and $v$: differentiate $u$ to get $du = 4 \cos(4x) dx$, and integrate $dv$ to get $v = -e^{-x}$.

Apply the integration by parts formula: substitute $u$, $v$, $du$ into $\int u \, dv = uv - \int v \, du$, then simplify the resulting integral. You may need to apply integration by parts a second time or solve for the original integral algebraically.

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Integration by Parts

Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals using the formula ∫u dv = uv - ∫v du. Choosing appropriate u and dv is crucial to simplify the integral effectively.

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