Evaluate the integral of (e^3x)(cosh)(2x)(dx) A.(1/2)(e^5x)+(1/2)(e^x)+C...


Evaluate the integral of {eq}(e^3x)(cosh)(2x)(dx) {/eq}

{eq}A.(1/2)(e^5x)+(1/2)(e^x)+C \\ B.(1/10)(e^5x)+(1/2)(e^x)+C \\ C.(1/4)(e^3x)+(1/2)(x)+C \\ D.(1/10)(e^5x)+(1/5)(x)+C\\ {/eq}

Integration of trigonometric function:

The above question concerns the topic of integration of trigonometric functions. In this question first we replace the hyperbolic function in terms of exponential function and then use the following formula to evaluate the integral.

{eq}\displaystyle \int e^{ax}dx=\frac{e^{ax}}{a}+C {/eq}

Answer and Explanation:

{eq}\begin{align} \displaystyle \int (e^3x)(cosh)(2x)(dx)\\ \text{we know that }cosh(2x)=\frac{e^{2x}+e^{-2x}}{2} \text{ Replacing this in integral, we have }\\ \displaystyle \int (e^{3x})\left (\frac{e^{2x}+e^{-2x}}{2} \right )dx\\ \displaystyle \Rightarrow \int \left (\frac{e^{5x}+e^{x}}{2} \right )\\ \displaystyle \Rightarrow \frac{1}{2}\left (\frac{e^{5x}}{5}+e^{x} \right )+C\\ \displaystyle \Rightarrow \frac{e^{5x}}{10}+\frac{e^{x}}{2} +C \end{align} {/eq}

so option B is correct

Learn more about this topic:

Integration Problems in Calculus: Solutions & Examples

from AP Calculus AB & BC: Homework Help Resource

Chapter 13 / Lesson 13

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