Find the following integral in exercise: e^xloga e^alogx e^aloga?

Question

Find the following integral in exercise: e^xloga   e^alogx   e^aloga?

Answer

Answer

Integrals of Exponential Functions

Integration of e^xloga

Let's consider the integral of e^xloga. We can rewrite this as:

e^xloga = (e^loga)^x = a^x

Therefore, the integral of e^xloga is:

∫ e^xloga dx = ∫ a^x dx = (a^x / ln a) + C

Integration of e^alogx

Next, we'll look at the integral of e^alogx. We can rewrite this as:

e^alogx = (e^a)^logx = x^a

Therefore, the integral of e^alogx is:

∫ e^alogx dx = ∫ x^a dx = (x^(a+1) / (a+1)) + C

Integration of e^aloga

Finally, we'll examine the integral of e^aloga. We can rewrite this as:

e^aloga = (e^a)^loga = a^a

Therefore, the integral of e^aloga is:

∫ e^aloga dx = ∫ a^a dx = (a^a / ln a) + C

Conclusion

Depending on the form of the exponential function, we can use different techniques to integrate it. By understanding the properties of exponential functions, we can simplify the integrals and find their antiderivatives.

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