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

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

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

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

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

Solution steps

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

Apply Trigonometric Substitution

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

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

Take the constant out:

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

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

Rewrite using trig identities

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

Expand

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

= 25 \cdot \int - sec (v) + sec^{3} (v) d v

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 (v) d v + \int sec^{3} (v) d v)

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

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

= 25 (- ln ∣ tan (v) + sec (v) ∣ + \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

Substitute back

v = arctan (\frac{1}{5} u)

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

Simplify

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

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

Add a constant to the solution

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

\frac{3}{5} x^{10}

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\int 6^{3} + \frac{3}{x} + \frac{1}{x ^{2}} d x

\frac{\partial}{\partial x} (4 y e^{5 x^{2} - 6 y})

x \to 0 + lim (x^{\frac{1}{x}})

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