Set up integral: Let's start by setting up the integral we need to solve: int(4x cos(2-3x)dx). We'll use a substitution method here. Let u = 2 - 3x, then du = -3 dx, or dx = -du/3. Substitute u and dx: Substitute u and dx in the integral: int(4x cos(u) * (-1/3) du). This simplifies to -4/3 int(x cos(u) du). Now, we need to express x in terms of u. From u = 2 - 3x, we solve for x: x = (2 - u)/3. Express x in terms of u: Plug x back into the integral: -4/3 int((2-u)/3 cos(u) du). Simplify the constants: -4/9 int((2-u) cos(u) du). Now, we need to distribute the cos(u) inside the integral: -4/9 (int(2cos(u) du) - int(u cos(u) du)). Distribute cos(u) inside integral: Solve the first integral int(2cos(u) du). This is straightforward: 2sin(u). For the second integral, int(u cos(u) du), use integration by parts. Let v = u, dv = cos(u) du, then du = dv, and v = sin(u).

Calculus

Find indefinite integrals using the substitution and by parts

Full solution

\int 4 x \cos (2-3 x) d x

Set up integral: Let's start by setting up the integral we need to solve: $\int 4x \cos(2-3x)\,dx$ . We'll use a substitution method here. Let $u = 2 - 3x$ , then $du = -3 \,dx$ , or $dx = -\frac{du}{3}$ .
Substitute $u$ and $dx$ : Substitute $u$ and $dx$ in the integral: $\int(4x \cos(u) * (-\frac{1}{3}) \,du)$ . This simplifies to $-\frac{4}{3} \int(x \cos(u) \,du)$ . Now, we need to express $x$ in terms of $u$ . From $u = 2 - 3x$ , we solve for $x$ : $dx$ $0$ .
Express $x$ in terms of $u$ : Plug $x$ back into the integral: $-\frac{4}{3} \int\left(\frac{2-u}{3} \cos(u) \,du\right)$ . Simplify the constants: $-\frac{4}{9} \int((2-u) \cos(u) \,du)$ . Now, we need to distribute the $\cos(u)$ inside the integral: $-\frac{4}{9} \left(\int(2\cos(u) \,du) - \int(u \cos(u) \,du)\right)$ .
Distribute $\cos(u)$ inside integral: Solve the first integral $\int(2\cos(u) \, du)$ . This is straightforward: $2\sin(u)$ . For the second integral, $\int(u \cos(u) \, du)$ , use integration by parts. Let $v = u$ , $dv = \cos(u) \, du$ , then $du = dv$ , and $v = \sin(u)$ .