What is the integral of 7x^3e^{5x} ?

The integral of 7x^3e^{5x} is 7/625 (125e^{5x}x^3-3(25e^{5x}x^2-2(5e^{5x}x-e^{5x})))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 7x^3e^{5x}

\int 7 x^{3} e^{5 x} d x

\frac{7}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x}))) + C

Solution steps

\int 7 x^{3} e^{5 x} d x

Take the constant out:

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

= 7 \cdot \int x^{3} e^{5 x} d x

Apply u-substitution

: \int \frac{e ^{u} u ^{3}}{625} d u

= 7 \cdot \int \frac{e ^{u} u ^{3}}{625} d u

Take the constant out:

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

= 7 \cdot \frac{1}{625} \cdot \int e^{u} u^{3} d u

Apply Integration By Parts

: u^{3} e^{u} - \int 3 u^{2} e^{u} d u

= 7 \cdot \frac{1}{625} (u^{3} e^{u} - \int 3 u^{2} e^{u} d u)

\int 3 u^{2} e^{u} d u = 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

= 7 \cdot \frac{1}{625} (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

Substitute back

u = 5 x

= 7 \cdot \frac{1}{625} ((5 x)^{3} e^{5 x} - 3 ((5 x)^{2} e^{5 x} - 2 (e^{5 x} \cdot 5 x - e^{5 x})))

Simplify

7 \cdot \frac{1}{625} ((5 x)^{3} e^{5 x} - 3 ((5 x)^{2} e^{5 x} - 2 (e^{5 x} \cdot 5 x - e^{5 x}))) : \frac{7}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x})))

= \frac{7}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x})))

Add a constant to the solution

= \frac{7}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x}))) + C

(1 + x)^{2}

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

n = 0 \sum \infty \frac{2 ^{n}}{5 ^{n}}

x \to \infty - lim (\frac{1}{x})

x y^{^{'}} = x e^{- 2 x} - (2 x + 1) y

What is the integral of 7x^3e^{5x} ?
The integral of 7x^3e^{5x} is 7/625 (125e^{5x}x^3-3(25e^{5x}x^2-2(5e^{5x}x-e^{5x})))+C