求 $\frac{4}{x^2} - \frac{3}{1 - 5x} + \sin 5x - \pi$ 关于 $x$ 的积分。

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完整解法

1. Rewrite the expression in a form ready for integration

$\int \left( 4x^{-2} - \frac{3}{1 - 5x} + \sin 5x - \pi \right) dx$

2. Integrate the power and logarithmic terms

$\int 4x^{-2} dx = \frac{4x^{-1}}{-1} = -\frac{4}{x}$
$\int -\frac{3}{1 - 5x} dx = -3 \left( \frac{\ln(1 - 5x)}{-5} \right) = \frac{3}{5}\ln(1 - 5x)$

3. Integrate the trigonometric and constant terms, and add the constant of integration

$\int \sin 5x dx = -\frac{1}{5}\cos 5x$
$\int -\pi dx = -\pi x$

Result: $-\frac{4}{x} + \frac{3}{5}\ln(1 - 5x) - \frac{1}{5}\cos 5x - \pi x + C$

答案

$-\frac{4}{x} + \frac{3}{5}\ln(1 - 5x) - \frac{1}{5}\cos 5x - \pi x + C$

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