Question:

Evaluate : $\int\left(e^{x\,log\,a} + e^{a\,log\,x} + e^{a\,log\,a} \right)dx$

Updated On: Jul 6, 2022
  • $\frac{a^{x}}{log \,a}+\frac{x^{a+1}}{a+1} + a^{a}x + c$
  • $\frac{a^{x}}{log \,a}+\frac{x^{a+1}}{a-1} + a x^{a} + c$
  • $\frac{a^{x}}{log\, a}+\frac{x^{a+1}}{a+1} + a x^{a} + c$
  • $\frac{a^{x}}{log \,a}-\frac{x^{a+1}}{a+1} + a^{a} x + c$
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The Correct Option is A

Solution and Explanation

We have,$ I =\int\left(e^{x\,log\,a}+e^{a\,log\,x} +e^{a\,log\,a}\right) dx $ Then,$ I =\int \left(e^{log\,a^x} +e^{log\,x^a} +e^{log\,a^a}\right)dx$ $\Rightarrow I = \int\left(a^{x} +x^{a} +a^{a}\right)dx \left[\because e^{log\lambda}=\lambda\right]$ $\Rightarrow\int a^{x}dx+\int x^{a}dx + \int a^{a}dx$ $\Rightarrow I =\frac{a^{x}}{loga}+\frac{x^{a+1}}{a+1} + a^{a } x +C$
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Notes on integral

Concepts Used:

Integral

The representation of the area of a region under a curve is called to be as integral. The actual value of an integral can be acquired (approximately) by drawing rectangles.

  • The definite integral of a function can be shown as the area of the region bounded by its graph of the given function between two points in the line.
  • The area of a region is found by splitting it into thin vertical rectangles and applying the lower and the upper limits, the area of the region is summarized.
  • An integral of a function over an interval on which the integral is described.

Also, F(x) is known to be a Newton-Leibnitz integral or antiderivative or primitive of a function f(x) on an interval I.

F'(x) = f(x)

For every value of x = I.

Types of Integrals:

Integral calculus helps to resolve two major types of problems:

  1. The problem of getting a function if its derivative is given.
  2. The problem of getting the area bounded by the graph of a function under given situations.