What is the integral of x/(sqrt(1+x^2+y^2)) ?

The integral of x/(sqrt(1+x^2+y^2)) is sqrt(1+y^2+x^2)+C

Popular >

integral of x/(sqrt(1+x^2+y^2))

\int \frac{x}{1 + x ^{2} + y ^{2}} d x

1 + y^{2} + x^{2} + C

Solution steps

\int \frac{x}{1 + x ^{2} + y ^{2}} d x

Apply u-substitution

: \int \frac{1}{2 u} d u

= \int \frac{1}{2 u} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= \frac{1}{2} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 1 + y^{2} + x^{2}

= \frac{1}{2} \cdot 2 (1 + y^{2} + x^{2})^{\frac{1}{2}}

Simplify

\frac{1}{2} \cdot 2 (1 + y^{2} + x^{2})^{\frac{1}{2}} : 1 + y^{2} + x^{2}

= 1 + y^{2} + x^{2}

Add a constant to the solution

= 1 + y^{2} + x^{2} + C

\int \frac{8 x ^{2} + x + 9}{( x + 1 ) ( x ^{2} + 1 )} d x

\int \frac{9}{3 e ^{- 2 v} - 7} d v

\int \frac{e ^{2 x}}{1 + 6 e ^{2 x} + 9 e ^{4 x}} d x

\int sin (7 x) cos (8 x) d x

\int \frac{ln ( 3 x )}{x ^{11}} d x

What is the integral of x/(sqrt(1+x^2+y^2)) ?
The integral of x/(sqrt(1+x^2+y^2)) is sqrt(1+y^2+x^2)+C