Question
upstudy study bank question image url

Hallar la integral indefinida. \[ \int(u+4)(2 u+1) d(u) \]

Ask by Marsh Lindsey. in Colombia
Mar 16,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

La integral indefinida es: \[ \frac{2u^3}{3} + \frac{9u^2}{2} + 4u + C. \]

Solution

1. Expandimos el producto de los polinomios: \[ (u+4)(2u+1) = u\cdot2u + u\cdot1 + 4\cdot2u + 4\cdot1 = 2u^2 + u + 8u + 4 = 2u^2 + 9u + 4. \] 2. Escribimos la integral con la expresión expandida: \[ \int (u+4)(2u+1) \, du = \int \left(2u^2 + 9u + 4\right) \, du. \] 3. Integramos término a término: \[ \int 2u^2 \, du = 2 \cdot \frac{u^3}{3} = \frac{2u^3}{3}, \] \[ \int 9u \, du = 9 \cdot \frac{u^2}{2} = \frac{9u^2}{2}, \] \[ \int 4 \, du = 4u. \] 4. Sumamos los resultados y añadimos la constante de integración \( C \): \[ \frac{2u^3}{3} + \frac{9u^2}{2} + 4u + C. \] La integral indefinida es: \[ \int(u+4)(2u+1) \, du = \frac{2u^3}{3} + \frac{9u^2}{2} + 4u + C. \]

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

The Deep Dive

Para resolver la integral indefinida \(\int(u+4)(2u+1) \, d(u)\), primero expandimos la expresión dentro de la integral: \[(u+4)(2u+1) = u(2u) + u(1) + 4(2u) + 4(1) = 2u^2 + u + 8u + 4 = 2u^2 + 9u + 4.\] Ahora, podemos escribir la integral: \[\int(2u^2 + 9u + 4) \, d(u).\] Procedemos a integrar término a término: 1. \(\int 2u^2 \, d(u) = \frac{2}{3}u^3\), 2. \(\int 9u \, d(u) = \frac{9}{2}u^2\), 3. \(\int 4 \, d(u) = 4u\). Así que sumando todo: \[\int(2u^2 + 9u + 4) \, d(u) = \frac{2}{3}u^3 + \frac{9}{2}u^2 + 4u + C,\] donde \(C\) es la constante de integración. Por lo tanto, la integral indefinida es: \[\frac{2}{3}u^3 + \frac{9}{2}u^2 + 4u + C.\]

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy