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int5x cos(5x)dx

$\int 5 x \cos (5 x) d x$

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Find indefinite integrals using the substitution and by parts

Full solution

Q.

\int 5 x \cos (5 x) d x

Recognize functions for integration: Recognize the integral as a product of two functions suitable for integration by parts. $\newline$ Integration by parts formula: $\int u \, dv = uv - \int v \, du$ $\newline$ Let $u = 5x$ and $dv = \cos(5x)\,dx$ .
Differentiate and integrate: Differentiate $u$ and integrate $dv$ . $\newline$ Differentiating $u$ gives us $du = 5dx$ . $\newline$ Integrating $dv$ gives us $v = \int \cos(5x)dx = \frac{1}{5}\sin(5x)$ .
Apply integration by parts: Apply the integration by parts formula. $\newline$ $\int 5x \cos(5x)\,dx = uv - \int v\,du$ $\newline$ = $(5x)(\frac{1}{5})\sin(5x) - \int(\frac{1}{5})\sin(5x)(5\,dx)$ $\newline$ = $x \sin(5x) - \int \sin(5x)\,dx$
Integrate $\sin(5x)$ : Integrate $\int \sin(5x)\,dx$ . $\int \sin(5x)\,dx = -\left(\frac{1}{5}\right)\cos(5x) + C$ , where $C$ is the constant of integration.
Combine final answer: Combine the results to get the final answer. $\newline$ $\int 5x \cos(5x)\,dx = x \sin(5x) - \left(-\left(\frac{1}{5}\right)\cos(5x)\right) + C$ $\newline$ $= x \sin(5x) + \left(\frac{1}{5}\right)\cos(5x) + C$

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