Question
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Answer
The integral of \(4^{x} e^{2x}\) with respect to \(x\) is \(\frac{(4e^{2})^{x}}{2\ln{(2e)}} + C\).
Solution
Calculate the integral \( \int 4^{x} e^{2x} dx \).
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
\(\int 4^{x}e^{2x} dx\)
- step1: Multiply the terms:
\(\int \left(4e^{2}\right)^{x} dx\)
- step2: Evaluate the integral:
\(\frac{\left(4e^{2}\right)^{x}}{\ln{\left(4e^{2}\right)}}\)
- step3: Simplify:
\(\frac{\left(4e^{2}\right)^{x}}{2\ln{\left(2e\right)}}\)
- step4: Add the constant of integral C:
\(\frac{\left(4e^{2}\right)^{x}}{2\ln{\left(2e\right)}} + C, C \in \mathbb{R}\)
The integral of \(4^{x} e^{2x}\) with respect to \(x\) is \(\frac{(4e^{2})^{x}}{2\ln{(2e)}} + C\), where \(C\) is an arbitrary constant.
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Simplify this solution The Deep Dive
To solve the integral \( \int 4^{x} e^{2 x} d x \), we can rewrite \( 4^x \) as \( (e^{\ln 4})^x = e^{x \ln 4} \). Therefore, we have: \[ \int 4^{x} e^{2 x} d x = \int e^{x \ln 4} e^{2 x} d x = \int e^{x(\ln 4 + 2)} d x. \] Now we can use substitution to solve this integral. Let \( u = x(\ln 4 + 2) \), so \( du = (\ln 4 + 2) dx \) or \( dx = \frac{du}{\ln 4 + 2} \). Thus, the integral becomes: \[ \int e^{u} \frac{du}{\ln 4 + 2} = \frac{1}{\ln 4 + 2} e^{u} + C = \frac{1}{\ln 4 + 2} e^{x(\ln 4 + 2)} + C. \] Substituting back gives the final result: \[ \int 4^{x} e^{2 x} d x = \frac{1}{\ln 4 + 2} e^{x(\ln 4 + 2)} + C. \]
