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Question: What is the integral of \(2{{e}^{2x}}\) ?...

What is the integral of 2e2x2{{e}^{2x}} ?

Explanation

Solution

To find the integral of 2e2x2{{e}^{2x}} , firstly, we will write it in mathematical form. Then, we will take the constant outside the integration. Then, we have to apply the chain rule since the function is not in its standard form ex{{e}^{x}} . That is, along with the result we have to divide it by the differentiation of the power of e. Now, we have to solve the required answer.

Complete step by step solution:
We have to find the integral of 2e2x2{{e}^{2x}} . Let us express this mathematically.
⇒∫2e2xdx\Rightarrow \int{2{{e}^{2x}}}dx
We know that ∫af(x)dx=a∫f(x)dx\int{af\left( x \right)}dx=a\int{f\left( x \right)}dx . Let us take the constant 2 outside.
⇒∫2e2xdx=2∫e2xdx\Rightarrow \int{2{{e}^{2x}}}dx=2\int{{{e}^{2x}}}dx
Let us integrate the above result. We know that ∫exdx=ex+C\int{{{e}^{x}}dx}={{e}^{x}}+C . We will have to use the chain rule since the function is not in its standard form ex{{e}^{x}} . That is, along with the result we have to divide it by the differentiation of the power of e.
⇒∫2e2xdx=(2e2x×1ddx(2x))+C\Rightarrow \int{2{{e}^{2x}}}dx=\left( 2{{e}^{2x}}\times \dfrac{1}{\dfrac{d}{dx}\left( 2x \right)} \right)+C
We know that ddx(ax)=addx(x)\dfrac{d}{dx}\left( ax \right)=a\dfrac{d}{dx}\left( x \right) . Therefore, the above equation becomes
⇒∫2e2xdx=(2e2x×12ddx(x))+C\Rightarrow \int{2{{e}^{2x}}}dx=\left( 2{{e}^{2x}}\times \dfrac{1}{2\dfrac{d}{dx}\left( x \right)} \right)+C
We know that ddx(xn)=nxn−1\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}} . Therefore, the above equation becomes
⇒∫2e2xdx=(2e2x×12×1)+C ⇒∫2e2xdx=(2e2x×12)+C \begin{aligned} & \Rightarrow \int{2{{e}^{2x}}}dx=\left( 2{{e}^{2x}}\times \dfrac{1}{2\times 1} \right)+C \\\ & \Rightarrow \int{2{{e}^{2x}}}dx=\left( 2{{e}^{2x}}\times \dfrac{1}{2} \right)+C \\\ \end{aligned}
Let us cancel the common factor 2.
⇒∫2e2xdx=(\requirecancel2e2x×1\requirecancel2)+C\Rightarrow \int{2{{e}^{2x}}}dx=\left( \require{cancel}\cancel{2}{{e}^{2x}}\times \dfrac{1}{\require{cancel}\cancel{2}} \right)+C
We can write the result of the above simplification as
⇒∫2e2xdx=e2x+C\Rightarrow \int{2{{e}^{2x}}}dx={{e}^{2x}}+C
Hence, the integral of 2e2x2{{e}^{2x}} is e2x+C{{e}^{2x}}+C.

Note: Students must know the integrals and differentiations of basic functions. They must never forget to add the constant after the integration. They must know to integrate functions using chain rule, substitution and integration by parts. They must always take the constant term in the product of the function outside the integral.