Integral E 4x Solved: The Pattern Students Should Notice
Integral of e^4x
The integral of e^4x is $$\frac{1}{4}e^{4x}+C$$, because the exponent's derivative is 4, so the antiderivative must be scaled by $$\frac{1}{4}$$ to reverse that factor.
Why the pattern works
This is one of the most useful exponential integration patterns in calculus: when you integrate $$e^{ax}$$, the result is $$\frac{1}{a}e^{ax}+C$$, provided $$a \neq 0$$. The reason is that differentiating $$e^{ax}$$ brings down the factor $$a$$, so integration must cancel it.
For exponential functions with a linear exponent, the constant in the exponent becomes a divisor in the answer.
Step-by-step solution
- Set $$u=4x$$, so the integral becomes $$\int e^u \cdot \frac{1}{4}\,du$$ after substitution.
- Pull out the constant $$\frac{1}{4}$$, giving $$\frac{1}{4}\int e^u\,du$$.
- Integrate $$e^u$$ to get $$\frac{1}{4}e^u+C$$.
- Substitute back $$u=4x$$, yielding $$\frac{1}{4}e^{4x}+C$$.
Formula table
| Integrand | Antiderivative | Rule |
|---|---|---|
| $$e^x$$ | $$e^x+C$$ | Basic exponential rule |
| $$e^{4x}$$ | $$\frac{1}{4}e^{4x}+C$$ | Divide by the exponent coefficient |
| $$e^{-4x}$$ | $$-\frac{1}{4}e^{-4x}+C$$ | Negative coefficient changes the sign |
Common student mistake
A frequent error is writing $$e^{4x}+C$$ without the $$\frac{1}{4}$$. That answer differentiates to $$4e^{4x}$$, not $$e^{4x}$$, so it is too large by a factor of 4.
Quick check
Differentiate $$\frac{1}{4}e^{4x}$$: the derivative is $$\frac{1}{4}\cdot 4e^{4x}=e^{4x}$$, which confirms the antiderivative is correct.
- $$\int e^{4x}\,dx=\frac{1}{4}e^{4x}+C$$.
- $$\int e^{ax}\,dx=\frac{1}{a}e^{ax}+C$$, for $$a\neq 0$$.
- The substitution $$u=4x$$ is the standard method when you want to show the rule step by step.
FAQ
What are the most common questions about Integral E 4x Solved The Pattern Students Should Notice?
What is the integral of e^4x?
$$\int e^{4x}\,dx=\frac{1}{4}e^{4x}+C$$.
Why do you divide by 4?
Because the derivative of $$4x$$ is 4, and integration reverses differentiation, so the answer must compensate by multiplying by $$\frac{1}{4}$$.
Does this rule work for any exponent ax?
Yes, for any constant $$a\neq 0$$, $$\int e^{ax}\,dx=\frac{1}{a}e^{ax}+C$$.
