Solving the integration problem:
$$\int \frac{4^x-5^x}{7^x} \, dx$$
Not sure how to solve this problem, I don't see a way to incorporate integration by parts into this problem or even some kind of substitution.
Seems that taking the natural log of each element will help to simplify the equation but I don't think it is valid mathematically.
HINT: $$\frac{4^x-5^x}{7^x}=\left(\frac47\right)^x-\left(\frac57\right)^x,$$ and you have the formula (by putting $a=4/7$ and $5/7$) $$\int a^x~\mathrm dx=?$$
$$I=\int\frac{a^x-b^x}{c^x}dx$$ $$I=\int\frac{a^x}{c^x}dx-\int\frac{b^x}{c^x}dx$$ $$I=\int \bigg(\frac ac\bigg)^xdx-\int\bigg(\frac bc\bigg)^xdx$$ Each integral is in the form $$G(k)=\int k^xdx$$ $$G(k)=\int e^{x\log k}dx$$ Substitute $u=x\log k$, which gives $$G(k)=\frac1{\log k}\int e^udu$$ $$G(k)=\frac1{\log k}e^{x\log k}$$ $$G(k)=\frac1{\log k}k^{x}+C$$ Thus $$I=G\bigg(\frac ac\bigg)-G\bigg(\frac bc\bigg)+C$$
Keep in mind $\log(\cdot)$ denotes the natural logarithm.
