Question

consider the indefinite integral $\int \frac{8 e^{8 x}}{\left(e^{8 x}+5\
ight)^{6}} d x$: this can be transformed into a basic integral by letting $u=$ and $d u=d x$ performing the substitution yields the integral $\int d u$

Explanation:

Step1: Set substitution

Let \(u = e^{8x}+5\).

Step2: Differentiate \(u\)

Differentiate \(u\) with respect to \(x\). Using the chain - rule \(\frac{du}{dx}=\frac{d}{dx}(e^{8x}+5)\). Since \(\frac{d}{dx}(e^{ax}) = ae^{ax}\) and \(\frac{d}{dx}(c)=0\) (where \(a = 8\) and \(c = 5\)), we have \(\frac{du}{dx}=8e^{8x}\), so \(du = 8e^{8x}dx\).

Step3: Substitute into the integral

Substitute \(u = e^{8x}+5\) and \(du = 8e^{8x}dx\) into the integral \(\int\frac{8e^{8x}}{(e^{8x}+5)^{6}}dx\). The integral becomes \(\int\frac{du}{u^{6}}\).

Answer:

\(u = e^{8x}+5\), \(du=8e^{8x}dx\), \(\int\frac{1}{u^{6}}du\)