Evaluate the integral. int-5x cos(3x)dx Answer:

Choose u and dv: To solve the integral of -5x cos(3x) with respect to x, we will use integration by parts, which is given by the formula ∫u dv = uv - ∫v du. We need to choose u and dv such that the resulting integral is simpler to solve. Let's choose u = -5x (which gives us du = -5 dx) and dv = cos(3x) dx (which gives us v = (1/3)sin(3x) after integration).Apply integration by parts: Now we apply the integration by parts formula: ∫-5x cos(3x) dx = uv - ∫v du = (-5x)(1/3)sin(3x) - ∫(1/3)sin(3x)(-5) dx = (-5/3)x sin(3x) + (5/3)∫sin(3x) dxIntegrate sin(3x): Next, we integrate (5/3)∫sin(3x) dx. The integral of sin(3x) with respect to x is (-1/3)cos(3x), so we have: (5/3)∫sin(3x) dx = (5/3)(-1/3)cos(3x) + C = -(5/9)cos(3x) + C, where C is the constant of integration.Combine calculated parts: Combining the two parts we have calculated, we get the final answer for the integral: ∫-5x cos(3x) dx = (-5/3)x sin(3x) - (5/9)cos(3x) + C

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Evaluate the integral.

int-5x cos(3x)dx
Answer:

Evaluate the integral. $\newline$ $\int-5 x \cos (3 x) d x$ $\newline$ Answer:

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Find indefinite integrals using the substitution and by parts

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Q. Evaluate the integral.

\newline

\int-5 x \cos (3 x) d x

\newline

Answer:

Choose $u$ and $dv$ : To solve the integral of $-5x \cos(3x)$ with respect to $x$ , we will use integration by parts, which is given by the formula $\int u \, dv = uv - \int v \, du$ . We need to choose $u$ and $dv$ such that the resulting integral is simpler to solve. $\newline$ Let's choose $u = -5x$ (which gives us $du = -5 \, dx$ ) and $dv = \cos(3x) \, dx$ (which gives us $dv$ $0$ after integration).
Apply integration by parts: Now we apply the integration by parts formula: $\newline$ $\int -5x \cos(3x) \, dx = uv - \int v \, du$ $\newline$ = $(-5x)(\frac{1}{3})\sin(3x) - \int(\frac{1}{3})\sin(3x)(-5) \, dx$ $\newline$ = $(-\frac{5}{3})x \sin(3x) + (\frac{5}{3})\int\sin(3x) \, dx$
Integrate $\sin(3x)$ : Next, we integrate $(5/3)\int \sin(3x) \, dx$ . The integral of $\sin(3x)$ with respect to $x$ is $(-1/3)\cos(3x)$ , so we have: $\newline$ $(5/3)\int \sin(3x) \, dx = (5/3)(-1/3)\cos(3x) + C = -(5/9)\cos(3x) + C$ , where $C$ is the constant of integration.
Combine calculated parts: Combining the two parts we have calculated, we get the final answer for the integral: $\newline$ \int \(-5x \cos( $3$ x) \, dx = \left(-\frac{ $5$ }{ $3$ }\right)x \sin( $3$ x) - \left(\frac{ $5$ }{ $9$ }\right)\cos( $3$ x) + C