QuestionAugust 25, 2025

Find an antiderivative of the function f(x)=7e^x e^7x (1)/(7)e^7x 7e^x (1)/(7)e^x

Find an antiderivative of the function f(x)=7e^x e^7x (1)/(7)e^7x 7e^x (1)/(7)e^x
Find an antiderivative of the function f(x)=7e^x e^7x (1)/(7)e^7x 7e^x (1)/(7)e^x

Solution
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Answer

7e^x + \frac{1}{49}e^{7x} + C Explanation 1. Identify the function components The function f(x) = 7e^x is a sum of two terms: 7e^x and \frac{1}{7}e^{7x}. 2. Find antiderivative of each term The antiderivative of 7e^x is 7 \int e^x \, dx = 7e^x + C_1. The antiderivative of \frac{1}{7}e^{7x} is \frac{1}{7} \int e^{7x} \, dx = \frac{1}{49}e^{7x} + C_2. 3. Combine results Combine the antiderivatives: 7e^x + \frac{1}{49}e^{7x} + C, where C = C_1 + C_2.

Explanation

1. Identify the function components<br /> The function $f(x) = 7e^x$ is a sum of two terms: $7e^x$ and $\frac{1}{7}e^{7x}$.<br />2. Find antiderivative of each term<br /> The antiderivative of $7e^x$ is $7 \int e^x \, dx = 7e^x + C_1$.<br /> The antiderivative of $\frac{1}{7}e^{7x}$ is $\frac{1}{7} \int e^{7x} \, dx = \frac{1}{49}e^{7x} + C_2$.<br />3. Combine results<br /> Combine the antiderivatives: $7e^x + \frac{1}{49}e^{7x} + C$, where $C = C_1 + C_2$.
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