8.2. 5 ### Integration by Parts: Example Problem#### Problem StatementEvaluate the following integral using integration by parts:\[ \int 8x \, e^{9x} \, dx \]#### Solution ProcessUse the integration by parts formula so that the new integral is simpler than the original one. The integration by parts formula is:\[ \int u \, dv = uv - \int v \, du \]### Applying Integration by PartsGiven the integral:\[ \int 8x \, e^{9x} \, dx \]Let's choose:- \( u = 8x \)- \( dv = e^{9x} \, dx \)Now, we need to find \( du \) and \( v \):\[ du = 8 \, dx \]\[ v = \int e^{9x} \, dx = \frac{1}{9} e^{9x} \]Using the integration by parts formula:\[ \int 8x \, e^{9x} \, dx = u \cdot v - \int v \, du \]Substitute the expressions for \( u \), \( v \), and \( du \):\[ \int 8x \, e^{9x} \, dx = 8x \cdot \frac{1}{9} e^{9x} - \int \frac{1}{9} e^{9x} \cdot 8 \, dx \]Simplify the expression:\[ \int 8x \, e^{9x} \, dx = \frac{8}{9} x \, e^{9x} - \frac{8}{9} \int e^{9x} \, dx \]Evaluate the remaining integral:\[ \int e^{9x} \, dx = \frac{1}{9} e^{9x} \]Hence, the expression becomes:\[ \int 8x \, e^{9x} \, dx = \frac{8}{9} x \, e^{9x} - \frac{8}{9} \cdot \frac{1}{9} e^{9x} \]Simplify:\[ \int 8x \, e^{9x} \, dx = \frac{8}{9} x \, e^{9x} - \frac{8

Name: 8.2. 5 ### Integration by Parts: Example Problem#### Problem Stateme
Uploaded: 2024-03-20T07:06:41.000Z
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Description: 8.2. 5 ### Integration by Parts: Example Problem#### Problem StatementEvaluate the following integral using integration by parts:\[ \int 8x \, e^{9x} \, dx \]#### Solution ProcessUse the integration by parts formula so that the new integral is simp