What is the integral of sqrt(9+7x^2) ?

The integral of sqrt(9+7x^2) is 1/2 xsqrt(7x^2+9)+1/2*9/(sqrt(7))ln((|sqrt(7)x+sqrt(9+7x^2)|)/3)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sqrt(9+7x^2)

\int 9 + 7 x^{2} d x

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

Solution steps

\int 9 + 7 x^{2} d x

Apply Trigonometric Substitution

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

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

Take the constant out:

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

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

Apply Integral Reduction

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

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

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

Substitute back

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

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

Simplify

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

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

Add a constant to the solution

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

\int (1 + 3 x^{2}) (5 - x) d x

\int \frac{1}{x ( x + 3 ) ( x + 1 )} d x

\int x cos (x^{4}) d x

\int (e^{x} cos (e^{x})) d x

\int cot^{5} (7 x) d x

What is the integral of sqrt(9+7x^2) ?
The integral of sqrt(9+7x^2) is 1/2 xsqrt(7x^2+9)+1/2*9/(sqrt(7))ln((|sqrt(7)x+sqrt(9+7x^2)|)/3)+C