What is the integral of x^2sqrt(3+7x^2) ?

The integral of x^2sqrt(3+7x^2) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x^2sqrt(3+7x^2)

\int x^{2} 3 + 7 x^{2} d x

- \frac{3}{56} x 7 x^{2} + 3 - \frac{9}{56 7} ln \frac{7}{3} x + \frac{1}{3} (3 + 7 x^{2}) + \frac{1}{28} x (7 x^{2} + 3)^{\frac{3}{2}} + C

Solution steps

\int x^{2} 3 + 7 x^{2} d x

Apply Trigonometric Substitution

: \int \frac{9}{7 7} sec^{3} (u) tan^{2} (u) d u

= \int \frac{9}{7 7} sec^{3} (u) tan^{2} (u) d u

Take the constant out:

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

= \frac{9}{7 7} \cdot \int sec^{3} (u) tan^{2} (u) d u

Rewrite using trig identities

= \frac{9}{7 7} \cdot \int sec^{3} (u) (- 1 + sec^{2} (u)) d u

Expand

sec^{3} (u) (- 1 + sec^{2} (u)) : - sec^{3} (u) + sec^{5} (u)

= \frac{9}{7 7} \cdot \int - sec^{3} (u) + sec^{5} (u) d u

Apply the Sum Rule:

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

= \frac{9}{7 7} (- \int sec^{3} (u) d u + \int sec^{5} (u) d u)

\int sec^{3} (u) d u = \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

\int sec^{5} (u) d u = \frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

= \frac{9}{7 7} (- (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣) + \frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣))

Substitute back

u = arctan (\frac{7}{3} x)

= \frac{9}{7 7} - (\frac{1}{2} sec (arctan (\frac{7}{3} x)) tan (arctan (\frac{7}{3} x)) + \frac{1}{2} ln tan (arctan (\frac{7}{3} x)) + sec (arctan (\frac{7}{3} x))) + \frac{sec ^{4} ( arctan ( \frac{7}{3} x ) ) sin ( arctan ( \frac{7}{3} x ) )}{4} + \frac{3}{4} (\frac{1}{2} sec (arctan (\frac{7}{3} x)) tan (arctan (\frac{7}{3} x)) + \frac{1}{2} ln tan (arctan (\frac{7}{3} x)) + sec (arctan (\frac{7}{3} x)))

Simplify

\frac{9}{7 7} - (\frac{1}{2} sec (arctan (\frac{7}{3} x)) tan (arctan (\frac{7}{3} x)) + \frac{1}{2} ln tan (arctan (\frac{7}{3} x)) + sec (arctan (\frac{7}{3} x))) + \frac{sec ^{4} ( arctan ( \frac{7}{3} x ) ) sin ( arctan ( \frac{7}{3} x ) )}{4} + \frac{3}{4} (\frac{1}{2} sec (arctan (\frac{7}{3} x)) tan (arctan (\frac{7}{3} x)) + \frac{1}{2} ln tan (arctan (\frac{7}{3} x)) + sec (arctan (\frac{7}{3} x))) : - \frac{3}{56} x 7 x^{2} + 3 - \frac{9}{56 7} ln \frac{7}{3} x + \frac{1}{3} (3 + 7 x^{2}) + \frac{1}{28} x (7 x^{2} + 3)^{\frac{3}{2}}

= - \frac{3}{56} x 7 x^{2} + 3 - \frac{9}{56 7} ln \frac{7}{3} x + \frac{1}{3} (3 + 7 x^{2}) + \frac{1}{28} x (7 x^{2} + 3)^{\frac{3}{2}}

Add a constant to the solution

= - \frac{3}{56} x 7 x^{2} + 3 - \frac{9}{56 7} ln \frac{7}{3} x + \frac{1}{3} (3 + 7 x^{2}) + \frac{1}{28} x (7 x^{2} + 3)^{\frac{3}{2}} + C

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

\int x \cdot e^{x^{2} + 1} d x

(5 x + y e^{\frac{y}{x}}) d x - (x e^{\frac{y}{x}}) d y = 0, y (1) = 3

\int \frac{3 x}{1 - x ^{2}} d x

\frac{\partial}{\partial y} (x y e^{- 7 y})