What is the integral of sqrt(x^2+6x+25) ?

The integral of sqrt(x^2+6x+25) is 16(1/32 (x+3)sqrt(x^2+6x+25)+1/2 ln((|x+3+sqrt(x^2+6x+25)|)/4))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sqrt(x^2+6x+25)

\int x^{2} + 6 x + 25 d x

16 (\frac{1}{32} (x + 3) x^{2} + 6 x + 25 + \frac{1}{2} ln (\frac{x + 3 + x ^{2} + 6 x + 25}{4})) + C

Solution steps

\int x^{2} + 6 x + 25 d x

Complete the square

x^{2} + 6 x + 25 : (x + 3)^{2} + 16

= \int (x + 3)^{2} + 16 d x

Apply u-substitution

: \int u^{2} + 16 d u

= \int u^{2} + 16 d u

Apply Trigonometric Substitution

: \int 16 sec^{3} (v) d v

= \int 16 sec^{3} (v) d v

Take the constant out:

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

= 16 \cdot \int sec^{3} (v) d v

Apply Integral Reduction

= 16 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} \cdot \int sec (v) d v)

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

= 16 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

Substitute back

= 16 (\frac{sec ^{2} ( arctan ( \frac{1}{4} ( x + 3 ) ) ) sin ( arctan ( \frac{1}{4} ( x + 3 ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{4} (x + 3))) + sec (arctan (\frac{1}{4} (x + 3))))

Simplify

16 (\frac{sec ^{2} ( arctan ( \frac{1}{4} ( x + 3 ) ) ) sin ( arctan ( \frac{1}{4} ( x + 3 ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{4} (x + 3))) + sec (arctan (\frac{1}{4} (x + 3)))) : 16 (\frac{1}{32} (x + 3) x^{2} + 6 x + 25 + \frac{1}{2} ln (\frac{x + 3 + x ^{2} + 6 x + 25}{4}))

= 16 (\frac{1}{32} (x + 3) x^{2} + 6 x + 25 + \frac{1}{2} ln (\frac{x + 3 + x ^{2} + 6 x + 25}{4}))

Add a constant to the solution

= 16 (\frac{1}{32} (x + 3) x^{2} + 6 x + 25 + \frac{1}{2} ln (\frac{x + 3 + x ^{2} + 6 x + 25}{4})) + C

\int \frac{( y ^{4} - 8 )}{y ^{3} + 2 y ^{2}} d y

\int csc (1 - 2 x) cot (1 - 2 x) d x

\int \frac{b x + c}{x ^{2} - 8 x + 25} d x

\int \frac{15}{( 0.5 t + 1 ) ^{2}} d t

\int 5 x x - 1 d x

What is the integral of sqrt(x^2+6x+25) ?
The integral of sqrt(x^2+6x+25) is 16(1/32 (x+3)sqrt(x^2+6x+25)+1/2 ln((|x+3+sqrt(x^2+6x+25)|)/4))+C