What is the integral of (xe^{5x})/((5x+1)^2) ?

The integral of (xe^{5x})/((5x+1)^2) is (e^{5x})/(25(5x+1))+C

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integral of (xe^{5x})/((5x+1)^2)

\int \frac{x e ^{5 x}}{( 5 x + 1 ) ^{2}} d x

\frac{e ^{5 x}}{25 ( 5 x + 1 )} + C

Solution steps

\int \frac{x e ^{5 x}}{( 5 x + 1 ) ^{2}} d x

Apply u-substitution

: \int \frac{e ^{u} u}{25 ( u + 1 ) ^{2}} d u

= \int \frac{e ^{u} u}{25 ( u + 1 ) ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{25} \cdot \int \frac{e ^{u} u}{( u + 1 ) ^{2}} d u

Apply Integration By Parts

: - \frac{e ^{u} u}{u + 1} - \int - e^{u} d u

= \frac{1}{25} (- \frac{e ^{u} u}{u + 1} - \int - e^{u} d u)

\int - e^{u} d u = - e^{u}

= \frac{1}{25} (- \frac{e ^{u} u}{u + 1} - (- e^{u}))

Substitute back

u = 5 x

= \frac{1}{25} (- \frac{e ^{5 x} \cdot 5 x}{5 x + 1} - (- e^{5 x}))

Simplify

\frac{1}{25} (- \frac{e ^{5 x} \cdot 5 x}{5 x + 1} - (- e^{5 x})) : \frac{e ^{5 x}}{25 ( 5 x + 1 )}

= \frac{e ^{5 x}}{25 ( 5 x + 1 )}

Add a constant to the solution

= \frac{e ^{5 x}}{25 ( 5 x + 1 )} + C

x \to 0 lim (e^{\frac{∣ x ∣}{x}})

cos^{2} (x)

(x^{2} - y^{2}) (\frac{d y}{d x}) = 2 x y

\frac{d}{d x} (\frac{sin ( x )}{1 + x ^{2}})

\int sin (π t) cos (π t) d t

What is the integral of (xe^{5x})/((5x+1)^2) ?
The integral of (xe^{5x})/((5x+1)^2) is (e^{5x})/(25(5x+1))+C