Evalúa las siguientes integrales:
Pregunta 1(i). ![Rendered by QuickLaTeX.com \int\limits_1^4 f\left( x \right) dx, where\ f\left( x \right) = \begin{cases}4x + 3 & , & \text{if }1 \leq x \leq 2 \\3x + 5 & , & \text{if }2 \leq x \leq 4\end{cases}](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-a6dc946d884518179d256a6f11d6835f_l3.png)
Solución:
Tenemos,
yo =
yo =
Usando la propiedad aditiva, obtenemos
yo =
yo =
yo = 8 + 6 – 2 – 3 + 24 + 20 – 6 – 10
yo = 37
Pregunta 1(ii). ![Rendered by QuickLaTeX.com \int\limits_0^9 f\left( x \right) dx, where\ f\left( x \right)= \begin{cases}\sin x & , & 0 \leq x \leq \frac{\pi}{2} \\ 1 & , & \frac{\pi}{2} \leq x \leq 3 \\ e^{x - 3} & , & 3 \leq x \leq 9\end{cases}](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-d7e54b86b81588fd9d8d364921d4c316_l3.png)
Solución:
Tenemos,
yo =
yo =
Usando la propiedad aditiva, obtenemos
yo =
yo =
yo =
yo = 0 + 1 + 3 – π/2 + mi 6 – mi 0
yo = 3 – π/2 + mi 6
Pregunta 1(iii). ![Rendered by QuickLaTeX.com \int\limits_1^4 f\left( x \right) dx, where\ f\left( x \right) = \begin{cases}7x + 3 & , & \text{if }1 \leq x \leq 3 \\ 8x & , & \text{if }3 \leq x \leq 4\end{cases}](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-f31dd2686bbf3e378bbbc9e9e8dec230_l3.png)
Solución:
Tenemos,
yo =
yo =
Usando la propiedad aditiva, obtenemos
yo =
yo =
yo =
yo = 63/2 + 9 – 7/2 – 3 + 64 – 36
yo = 56/2 + 34
yo = 62
Pregunta 2. ![Rendered by QuickLaTeX.com \int\limits_{- 4}^4 \left| x + 2 \right| dx](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-56bef37431b359e183772fbbaf2e0658_l3.png)
Solución:
Tenemos,
yo =
Lo sabemos,
Entonces, obtenemos
yo =
yo =
yo =
yo = –2 + 4 – 8 – 8 + 8 + 8 – 2 + 4
yo = 20
Pregunta 3. ![Rendered by QuickLaTeX.com \int\limits_{- 3}^3 \left| x + 1 \right| dx](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-8e96baae8336ba231edaa4b7317b5209_l3.png)
Solución:
Tenemos,
yo =
Lo sabemos,
Entonces, obtenemos
yo =
yo =
yo = 0 + 2 + 8 – 0
yo = 10
Pregunta 4. ![Rendered by QuickLaTeX.com \int\limits_{- 1}^1 \left| 2x + 1 \right| dx](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-16877b3c9d72099f74ae324433470e46_l3.png)
Solución:
Tenemos,
yo =
Lo sabemos,
Entonces, obtenemos
yo =
yo =
yo = –1/4 + 1/2 + 1 – 1 + 1 + 1 – 1/4 + 1/2
yo = 5/2
Pregunta 5. ![Rendered by QuickLaTeX.com \int\limits_{- 2}^2 \left| 2x + 3 \right| dx](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-826f11cb7234b60be56a73b8caf78333_l3.png)
Solución:
Tenemos,
yo =
Lo sabemos,
Entonces, obtenemos
yo =
yo =
yo = –9/4 + 9/2 + 4 – 6 + 4 + 6 – 9/4 + 9/2
yo = 25/2
Pregunta 6. ![Rendered by QuickLaTeX.com \int\limits_0^2 \left| x^2 - 3x + 2 \right| dx](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-a0d537eca588db11c370fe215ac1a1a0_l3.png)
Solución:
Tenemos,
yo =
Lo sabemos,
Entonces obtenemos,
yo =
yo =
yo =
yo = 1/3 – 3/2 + 2 – [8/3 – 6 + 4 – 1/3 + 3/2 – 2]
yo = 1/3 – 3/2 + 2 – 8/3 + 6 – 2 + 1/3 – 3/2
yo = 1
Pregunta 7. ![Rendered by QuickLaTeX.com \int\limits_0^3 \left| 3x - 1 \right| dx](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-8df370f1f3580e312d676fb0b693953c_l3.png)
Solución:
Tenemos,
yo =
Lo sabemos,
Entonces obtenemos,
yo =
yo =
yo = –1/6 + 1/3 – 0 + 27/2 + 3 – 1/6 – 1/3
yo = 65/6
pregunta 8 ![Rendered by QuickLaTeX.com \int\limits_{- 6}^6 \left| x + 2 \right| dx](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-e00ec52badefefff765319a5b0f2702b_l3.png)
Solución:
Tenemos,
yo =
Lo sabemos,
Entonces, obtenemos
yo =
yo =
yo =
yo = –2 + 4 + 18 – 12 + 18 + 12 – 2 + 4
yo = 40
Pregunta 9. ![Rendered by QuickLaTeX.com \int\limits_{- 2}^2 \left| x + 1 \right| dx](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-9d560ca9a1de7f3551a5c8c0e5adb577_l3.png)
Solución:
Tenemos,
yo =
Lo sabemos,
Entonces obtenemos,
yo =
yo =
yo =
yo = -1/2 + 1 + 2 – 2 + 2 + 2 – 1/2 + 1
yo = 5
Pregunta 10. ![Rendered by QuickLaTeX.com \int\limits_1^2 \left| x - 3 \right| dx](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-921d3546d8b09ae93538085345c653f2_l3.png)
Solución:
Tenemos,
yo =
Lo sabemos,
Entonces obtenemos,
yo =
yo =
yo =
yo = – 2 – 6 + 1/2 + 3
yo = – 5 + 1/2
yo = (-10 + 1)/2
yo = -9/2
Pregunta 11. ![Rendered by QuickLaTeX.com \int\limits_0^{\pi/2} \left| \cos 2x \right| dx](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-278c7cae78adc5f63aed00c74d8b5f98_l3.png)
Solución:
Tenemos,
yo =
Lo sabemos,
Entonces obtenemos,
yo =
yo =
yo =
yo = 1/2 – 0 – 0 + 1/2
yo = 1
Pregunta 12. ![Rendered by QuickLaTeX.com \int\limits_0^{2\pi} \left| \sin x \right| dx](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-4385e12163ea34f2b7c38eaee7d12706_l3.png)
Solución:
Tenemos,
yo =
Lo sabemos,
Entonces obtenemos,
yo =
yo =
yo =
yo = 1 + 1 + 1 – (–1)
yo = 1 + 1 + 1 + 1
yo = 4
Pregunta 13. ![Rendered by QuickLaTeX.com \int\limits_{- \pi/4}^{\pi/4} \left| \sin x \right| dx](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-bf993534111a57dc59b127085054d45a_l3.png)
Solución:
Tenemos,
yo =
Lo sabemos,
Entonces obtenemos,
yo =
yo =
yo =
Yo = 1 – 1/√2 – 1/√2 + 1
Yo = 2 – 2/√2
yo = 2 – √2
Pregunta 14. ![Rendered by QuickLaTeX.com \int\limits_2^8 \left| x - 5 \right| dx](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-22c7d8e9ff8153eef3b10a075d4cc2fe_l3.png)
Solución:
Tenemos,
yo =
Lo sabemos,
Entonces obtenemos,
yo =
yo =
yo =
yo = – 25/2 + 25 + 2 – 10 + 32 – 40 – 25/2 + 25
yo = 9
Publicación traducida automáticamente
Artículo escrito por prabhjotkushparmar y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA