Find the integral of 4e^(2x) - sin(2x) with respect to x.

Step-by-step solution

1. 1

   Split the integral by linearity. Integration distributes over sums and differences and lets constants come out front:

   $\int\left(4e^{2x}-\sin 2x\right)dx=4\int e^{2x}\,dx-\int\sin 2x\,dx$

   Each piece is now a standard form with a linear inner function $2x$.

2. 2

   Remember the reversed chain rule for a linear inside. Differentiating $e^{2x}$ produces an extra factor $2$, so integrating must divide by it:

   $\int e^{ax}dx=\frac{1}{a}e^{ax}+C,\qquad \int\sin ax\,dx=-\frac{1}{a}\cos ax+C$

   This division is the step most often forgotten, and it changes the answer by a factor of two here.

3. 3

   Integrate the exponential term.

   $4\int e^{2x}dx=4\cdot\frac12 e^{2x}=2e^{2x}$

4. 4

   Integrate the sine term, tracking two minus signs. The rule contributes one minus and the original expression another:

   $-\int\sin 2x\,dx=-\left(-\frac12\cos 2x\right)=+\frac12\cos 2x$

   The result is $+\tfrac12\cos 2x$, not $-\tfrac12\cos 2x$.

5. 5

   Combine and add the constant of integration.

   $\int\left(4e^{2x}-\sin 2x\right)dx=2e^{2x}+\frac12\cos 2x+C$

6. 6

   Check by differentiating back. $\frac{d}{dx}\left(2e^{2x}\right)=4e^{2x}$ and $\frac{d}{dx}\left(\tfrac12\cos 2x\right)=-\sin 2x$, which reproduces the integrand exactly. Numerically at $x=0.6$ the derivative of the answer is $12.34843$ and the integrand is $12.34843$ ✓.