Answered

IDNStudy.com, ang iyong gabay para sa maaasahan at eksaktong mga sagot. Alamin ang mga maaasahang sagot sa iyong mga tanong mula sa aming malawak na kaalaman sa mga eksperto.

integrate this definite integral \[
\int_{1}^{3} \frac{2^{\log_3 (x^3 + 2x^2)}}{x \cdot 4^{\log_3 \sqrt{x+2}}} \, dx
\]

down below is the hint on how to solve it:

Integrate This Definite Integral Int13 Frac2log3 X3 2x2x Cdot 4log3 Sqrtx2 Dx Down Below Is The Hint On How To Solve It class=

Sagot :

Zjuidn

Use the hint to express the integral as shown below.

[tex]\begin{gathered} \int_{1}^{3} \frac{2^{\log_3 (x^3 + 2x^2)}}{x \cdot 4^{\log_3 \sqrt{x+2}}} \, dx = \int_{1}^{3} \frac{(x^3 + 2x^2)^{\log_3 2}}{x \cdot (\sqrt{x+2})^{\log_3 4}} \, dx \end{gathered}[/tex]

Remember that log₃(4) = 2log₃(2) and use the power of a power law of exponent to turn the latter integral as below.

[tex]\begin{gathered}\int_{1}^{3} \frac{(x^3 + 2x^2)^{\log_3 2}}{x \cdot (\sqrt{x+2})^{\log_3 4}} \, dx = \int_{1}^{3} \frac{1}{x}\left(\frac{x^3 + 2x^2}{(\sqrt{x+2})^{2}} \right)^{\log_3 2} \, dx \end{gathered}[/tex]

You can do the rest. I am sure you know how to simplify the fraction inside the parentheses. You should be able to get:

[tex]\begin{gathered} \int_{1}^{3} \frac{x^{2\log _3 2}}{x} \, dx = \int_{1}^{3} x^{\log _3 4-1}\, dx = \left.\frac{x^{\log _3 4}}{\log _3 4}\right|_{1}^{3}\end{gathered}[/tex]

This simplifies to 3/log₃(4) = log(27)/log(4).

Ang iyong kontribusyon ay napakahalaga sa amin. Magpatuloy sa pagtatanong at pagbabahagi ng iyong nalalaman. Sama-sama tayong magtatagumpay sa ating layunin. Para sa mga de-kalidad na sagot, piliin ang IDNStudy.com. Salamat at bumalik ka ulit sa aming site.