Arthur Pratt

2021-12-31

Is there a general formula for finding the primitive of

${e}^{ax}\mathrm{sin}\left(bx\right)$

Philip Williams

Beginner2022-01-01Added 39 answers

There is a pattern. Differentiating a function of the form ${e}^{ax}\mathrm{sin}\left(bx\right)$ yields a linear combination of a function of the same form, and a function ${e}^{ax}\mathrm{cos}\left(bx\right)$ . The analogous property holds for functions ${e}^{ax}\mathrm{cos}\left(bx\right)$ . So the primitive of ${e}^{ax}\mathrm{sin}\left(bx\right)$ will be a linear combination of ${e}^{ax}\mathrm{sin}\left(bx\right)$ and ${e}^{ax}\mathrm{cos}\left(bx\right)$ (plus a constant).

It remains to find the coefficients.

$\frac{d}{dx}({e}^{ax}(m\mathrm{sin}\left(bx\right)+n\mathrm{cos}\left(bx\right))={e}^{ax}(a(m\mathrm{sin}\left(bx\right)+n\left(\mathrm{cos}\left(bx\right)\right)+(bm\mathrm{cos}\left(bx\right)-bn\mathrm{sin}\left(bx\right)))$

$={e}^{ax}((am-bn)\mathrm{sin}\left(bx\right)+(an+bm)\mathrm{cos}\left(bx\right))$

Now solve the linear system

$am-bn=1$

$an+bm=0$

It remains to find the coefficients.

Now solve the linear system

Barbara Meeker

Beginner2022-01-02Added 38 answers

Hint Integration by parts

$\int udv=uv-\int vdu$

Make substition

$u=\mathrm{sin}\left(bx\right)\Rightarrow du=b\mathrm{cos}\left(bx\right)dx$

and

$dv={e}^{ax}\mathrm{sin}\left(bx\right)dx=\frac{{e}^{ax}}{a}\mathrm{sin}\left(bx\right)-\frac{b}{a}\int {e}^{ax}\mathrm{cos}bxdx$

Then another integration by parts for$\int {e}^{ax}\mathrm{cos}bxdx$ . I think you can do the rest of it.

Make substition

and

Then another integration by parts for

Vasquez

Expert2022-01-09Added 669 answers

Try by parts twice (assuming

and thus

Well, now just past the last rightmost summand to the left side (above) and do a little algebra:

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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