What is the integral of (x^2)/(sqrt(49+81x^2)) ?

The integral of (x^2)/(sqrt(49+81x^2)) is 1/1458 (-49ln((|9x+sqrt(81x^2+49)|)/7)+9xsqrt(81x^2+49))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^2)/(sqrt(49+81x^2))

\int \frac{x ^{2}}{49 + 81 x ^{2}} d x

\frac{1}{1458} (- 49 ln (\frac{9 x + 81 x ^{2} + 49}{7}) + 9 x 81 x^{2} + 49) + C

Solution steps

\int \frac{x ^{2}}{49 + 81 x ^{2}} d x

Apply Trigonometric Substitution

: \int \frac{49 tan ^{2} ( u ) sec ( u )}{729} d u

= \int \frac{49 tan ^{2} ( u ) sec ( u )}{729} d u

Take the constant out:

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

= \frac{49}{729} \cdot \int tan^{2} (u) sec (u) d u

Rewrite using trig identities

= \frac{49}{729} \cdot \int (- 1 + sec^{2} (u)) sec (u) d u

Expand

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

= \frac{49}{729} \cdot \int - sec (u) + sec^{3} (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{49}{729} (- \int sec (u) d u + \int sec^{3} (u) d u)

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

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

= \frac{49}{729} (- ln ∣ tan (u) + sec (u) ∣ + \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Substitute back

u = arctan (\frac{9}{7} x)

= \frac{49}{729} (- ln tan (arctan (\frac{9}{7} x)) + sec (arctan (\frac{9}{7} x)) + \frac{1}{2} sec (arctan (\frac{9}{7} x)) tan (arctan (\frac{9}{7} x)) + \frac{1}{2} ln tan (arctan (\frac{9}{7} x)) + sec (arctan (\frac{9}{7} x)))

Simplify

\frac{49}{729} (- ln tan (arctan (\frac{9}{7} x)) + sec (arctan (\frac{9}{7} x)) + \frac{1}{2} sec (arctan (\frac{9}{7} x)) tan (arctan (\frac{9}{7} x)) + \frac{1}{2} ln tan (arctan (\frac{9}{7} x)) + sec (arctan (\frac{9}{7} x))) : \frac{1}{1458} (- 49 ln (\frac{9 x + 81 x ^{2} + 49}{7}) + 9 x 81 x^{2} + 49)

= \frac{1}{1458} (- 49 ln (\frac{9 x + 81 x ^{2} + 49}{7}) + 9 x 81 x^{2} + 49)

Add a constant to the solution

= \frac{1}{1458} (- 49 ln (\frac{9 x + 81 x ^{2} + 49}{7}) + 9 x 81 x^{2} + 49) + C

f (x) = 3 x^{2} - 5 x + 7, (- 1, 15)

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

\int_{2}^{\infty} \frac{8}{x ^{3}} d x

\frac{1}{x - 1}

\int \frac{10}{x} + \frac{2}{x ^{2}} + \frac{7}{x ^{3}} d x

What is the integral of (x^2)/(sqrt(49+81x^2)) ?
The integral of (x^2)/(sqrt(49+81x^2)) is 1/1458 (-49ln((|9x+sqrt(81x^2+49)|)/7)+9xsqrt(81x^2+49))+C