Questions: ∫ dx/(3-8x) ; ∫ d/(x²-5x) ∫ dx/(5x+1)¹⁰ ; ∫(2x+4)

Transcript text: (10) \[ \begin{array}{l} \int \frac{d x}{3-8 x} ; \int \frac{d}{x^{2}-5 x} \\ \int \frac{d x}{(5 x+1)^{10}} ; \int(2 x+4) \end{array} \]

Solution

Solution Steps

Step 1: Integral of \(\int \frac{dx}{3-8x}\)

To solve the integral \(\int \frac{dx}{3-8x}\), we find that the result is: \[ \int \frac{dx}{3-8x} = -\frac{1}{8} \log(8x - 3) + C_1 \]

Step 2: Integral of \(\int \frac{dx}{x^2 - 5x}\)

For the integral \(\int \frac{dx}{x^2 - 5x}\), we factor the denominator and apply partial fraction decomposition. The result is: \[ \int \frac{dx}{x^2 - 5x} = -\frac{1}{5} \log|x| + \frac{1}{5} \log|x - 5| + C_2 \]

Step 3: Integral of \(\int \frac{dx}{(5x+1)^{10}}\)

For the integral \(\int \frac{dx}{(5x+1)^{10}}\), we use substitution and find: \[ \int \frac{dx}{(5x+1)^{10}} = -\frac{1}{87890625x^9 + 158203125x^8 + 126562500x^7 + 59062500x^6 + 17718750x^5 + 3543750x^4 + 472500x^3 + 40500x^2 + 2025x + 45} + C_3 \]

Final Answer

The results for the integrals are:

\(\int \frac{dx}{3-8x} = -\frac{1}{8} \log(8x - 3) + C_1\)
\(\int \frac{dx}{x^2 - 5x} = -\frac{1}{5} \log|x| + \frac{1}{5} \log|x - 5| + C_2\)
\(\int \frac{dx}{(5x+1)^{10}} = -\frac{1}{87890625x^9 + 158203125x^8 + 126562500x^7 + 59062500x^6 + 17718750x^5 + 3543750x^4 + 472500x^3 + 40500x^2 + 2025x + 45} + C_3\)

Thus, the final boxed answers are: \[ \boxed{\int \frac{dx}{3-8x} = -\frac{1}{8} \log(8x - 3) + C_1} \] \[ \boxed{\int \frac{dx}{x^2 - 5x} = -\frac{1}{5} \log|x| + \frac{1}{5} \log|x - 5| + C_2} \] \[ \boxed{\int \frac{dx}{(5x+1)^{10}} = -\frac{1}{87890625x^9 + 158203125x^8 + 126562500x^7 + 59062500x^6 + 17718750x^5 + 3543750x^4 + 472500x^3 + 40500x^2 + 2025x + 45} + C_3} \]

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