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

The integral of (x^2)/(sqrt(x^2+25)) is 25(-1/2 ln((|x+sqrt(25+x^2)|)/5)+1/50 xsqrt(25+x^2))+C

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

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

25 (- \frac{1}{2} ln (\frac{x + 25 + x ^{2}}{5}) + \frac{1}{50} x 25 + x^{2}) + C

Solution steps

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

Apply Trigonometric Substitution

: \int 25 tan^{2} (u) sec (u) d u

= \int 25 tan^{2} (u) sec (u) d u

Take the constant out:

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

= 25 \cdot \int tan^{2} (u) sec (u) d u

Rewrite using trig identities

= 25 \cdot \int (- 1 + sec^{2} (u)) sec (u) d u

Expand

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

= 25 \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

= 25 (- \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) ∣

= 25 (- 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{1}{5} x)

= 25 (- ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x)) + \frac{1}{2} sec (arctan (\frac{1}{5} x)) tan (arctan (\frac{1}{5} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x)))

Simplify

25 (- ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x)) + \frac{1}{2} sec (arctan (\frac{1}{5} x)) tan (arctan (\frac{1}{5} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x))) : 25 (- \frac{1}{2} ln (\frac{x + 25 + x ^{2}}{5}) + \frac{1}{50} x 25 + x^{2})

= 25 (- \frac{1}{2} ln (\frac{x + 25 + x ^{2}}{5}) + \frac{1}{50} x 25 + x^{2})

Add a constant to the solution

= 25 (- \frac{1}{2} ln (\frac{x + 25 + x ^{2}}{5}) + \frac{1}{50} x 25 + x^{2}) + C

y^{^{'}} = 7 y^{2}

\frac{d}{d x} (\frac{x}{3 2 x})

y^{^{''}} - 8 y^{^{'}} + 16 y = t^{- 6} e^{4 t}

\frac{\partial}{\partial y} (8 x^{5 y})

\frac{d y}{d t} = 2 y (1 - \frac{y}{2})

What is the integral of (x^2)/(sqrt(x^2+25)) ?
The integral of (x^2)/(sqrt(x^2+25)) is 25(-1/2 ln((|x+sqrt(25+x^2)|)/5)+1/50 xsqrt(25+x^2))+C