add question

questions/integrals/s25f-wk1.md

@@ -0,0 +1,48 @@
+---
+topic: integrals
+title: target practice
+desc: 
+date: 2025 January 19
+tags:
+  - fun
+  - exp
+method:
+  - work
+---
+
+
+## Question
+```math
+\int
+  2^x \, 4^x \, 8^x \cdots 256^x
+\ dx
+```
+
+
+## Hints
+
+### 1
+When we’re dealing with exponents, we often want them in terms of the same base.
+
+
+## Answer
+```math
+\frac{1}{36\ln2}2^{36x}-c
+```
+
+
+## Solution
+
+### Solve
+```math
+\begin{align*}
+  &\ \int 2^{x}\ 4^{x}\ 8^{x}\ ...\ 256^{x}\ dx
+  \\ =&\ \int 2^{x}\ 2^{2x}\ 2^{3x}\ ...\ 2^{8x}\ dx
+  \\ =&\ \int 2^{\left(x+2x+3x+...+8x\right)}\ dx
+  \\ =&\ \int 2^{36x}\ dx
+  \\ =&\ \int e^{\left(\ln2\right)\left(36x\right)}\ dx
+  \\ =&\ \int e^{\left(36\ln2\right)x}\ dx
+  \\ =&\ \frac{1}{36\ln2}e^{\left(36\ln2\right)x}
+  \\ =&\ \frac{1}{36\ln2}2^{36x}-c
+\end{align*}
+```