Integrate twice and solve for the repeated integral

\int e^{2x}\cos3x dx

Solution

Integral

Hint

Answer · Integral

e^{2x}(2\cos3x+3\sin3x)/13+C

Explore the idea

Drag x. The shading shows the selected interval; the lower curves compare the numerical signed integral with the change in the antiderivative.

Upper limit xx = 0.65

The limits run left to right. Area above the axis adds; area below the axis subtracts.

All three bars share one scale. Positive contributions extend right of zero; negative contributions extend left.

Signed integral from 0.00 to x

0.424

antiderivative − F(0)

0.424

agrees at displayed precision

Formula − integral

0.00% of the integral’s magnitude

Compare the running integral on a graph

Key idea

The two curves overlap exactly: the factor $\frac1{13}$ is essential.

Ideas to try

Check the dashed curve against the solid curve: they agree only with the factor

\frac1{13}