What is the integral of (3x+5)^2((7x-4)/(x^2)) ?

The integral of (3x+5)^2((7x-4)/(x^2)) is (63x^2)/2+174x+55ln|x|+(100)/x+C

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integral of (3x+5)^2((7x-4)/(x^2))

\int (3 x + 5)^{2} (\frac{7 x - 4}{x ^{2}}) d x

\frac{63 x ^{2}}{2} + 174 x + 55 ln ∣ x ∣ + \frac{100}{x} + C

Solution steps

\int (3 x + 5)^{2} \frac{7 x - 4}{x ^{2}} d x

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \int \frac{( 7 x - 4 ) ( 3 x + 5 ) ^{2}}{x ^{2}} d x

Expand

\frac{( 7 x - 4 ) ( 3 x + 5 ) ^{2}}{x ^{2}} : 63 x + 174 + \frac{55}{x} - \frac{100}{x ^{2}}

= \int 63 x + 174 + \frac{55}{x} - \frac{100}{x ^{2}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 63 x d x + \int 174 d x + \int \frac{55}{x} d x - \int \frac{100}{x ^{2}} d x

\int 63 x d x = \frac{63 x ^{2}}{2}

\int 174 d x = 174 x

\int \frac{55}{x} d x = 55 ln ∣ x ∣

\int \frac{100}{x ^{2}} d x = - \frac{100}{x}

= \frac{63 x ^{2}}{2} + 174 x + 55 ln ∣ x ∣ - (- \frac{100}{x})

Simplify

= \frac{63 x ^{2}}{2} + 174 x + 55 ln ∣ x ∣ + \frac{100}{x}

Add a constant to the solution

= \frac{63 x ^{2}}{2} + 174 x + 55 ln ∣ x ∣ + \frac{100}{x} + C

x \to + 0 lim (- \frac{7}{1 - 7 x})

f (x) = 7 x - 3

\frac{d y}{d x}, 5 x^{2} + 3 x y - y^{2} = 2

\frac{d}{d x} (x^{2 c o s (x)})

\frac{x}{1 + x}

What is the integral of (3x+5)^2((7x-4)/(x^2)) ?
The integral of (3x+5)^2((7x-4)/(x^2)) is (63x^2)/2+174x+55ln|x|+(100)/x+C