# How do you integrate #int (sinx)(5^x)# using integration by parts?

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The user is interested in math and specifically in integration by parts.To integrate ( \int \sin(x) \cdot 5^x ) using integration by parts, let's use the formula:

[ \int u , dv = uv - \int v , du ]

Let's choose ( u = \sin(x) ) and ( dv = 5^x , dx ). Then, we have ( du = \cos(x) , dx ) and ( v = \frac{1}{\ln(5)} \cdot 5^x ).

Now, we can apply the integration by parts formula:

[ \begin{aligned} \int \sin(x) \cdot 5^x , dx & = -\frac{1}{\ln(5)} \cdot \sin(x) \cdot 5^x - \int -\frac{1}{\ln(5)} \cdot \cos(x) \cdot 5^x , dx \ & = -\frac{1}{\ln(5)} \cdot \sin(x) \cdot 5^x + \frac{1}{\ln(5)} \cdot \int \cos(x) \cdot 5^x , dx \end{aligned} ]

We can now integrate ( \int \cos(x) \cdot 5^x ) using integration by parts again or use a different method, such as recognizing that ( \int \cos(x) \cdot 5^x ) is of the form ( \int \cos(x) \cdot a^x ), which can be integrated using a simple substitution. Would you like to see that?

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