- #1

mathnoob12345

- 1

- 0

teacher says the answer is: 2^(2x) / 2Ln(2)

why 2 times Ln(2)?

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In summary, the integral of 2^(2x) is (2^(2x))/(ln2), which can be found using the power rule of integration. No special formulas are needed and an example of solving for the integral is (2^(2x))/(ln2) + C. It is important to understand how to find the integral for the exam tomorrow as it is a fundamental concept in calculus.

- #1

mathnoob12345

- 1

- 0

teacher says the answer is: 2^(2x) / 2Ln(2)

why 2 times Ln(2)?

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- #2

MarkFL

Gold Member

MHB

- 13,288

- 12

\(\displaystyle 2^{2x}=e^{\ln\left(2^{2x}\right)}=e^{2\ln\left(2\right)x}\)

Hence:

\(\displaystyle I=\int 2^{2x}\,dx=\int e^{2\ln\left(2\right)x}\,dx\)

Let:

\(\displaystyle u=2\ln\left(2\right)x\implies du=2\ln\left(2\right)\,dx\)

And so we have:

\(\displaystyle I=\frac{1}{2\ln\left(2\right)}\int e^u\,du\)

Can you proceed?

The integral of 2^(2x) is equal to (2^(2x))/(ln2), where ln is the natural logarithm.

To solve for the integral of 2^(2x), you can use the power rule of integration, which states that the integral of x^n is equal to (x^(n+1))/(n+1). In this case, n = 2x, so the integral becomes (2^(2x))/(2x+1).

No, you do not need any special formulas to find the integral of 2^(2x). You can simply use the power rule of integration to solve for the integral.

Sure, for example, the integral of 2^(2x) dx is equal to (2^(2x))/(ln2) + C, where C is the constant of integration.

Yes, it is important to have a good understanding of how to find the integral of 2^(2x) for the exam tomorrow, as it is a fundamental concept in calculus and may be tested on the exam.

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