What is the integral of (a^2+b^2x^2)^{1/2}x ?

The integral of (a^2+b^2x^2)^{1/2}x is ((a^2+b^2x^2)^{3/2})/(3b^2)+C

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integral of (a^2+b^2x^2)^{1/2}x

\int (a^{2} + b^{2} x^{2})^{\frac{1}{2}} x d x

\frac{( a ^{2} + b ^{2} x ^{2} ) ^{\frac{3}{2}}}{3 b ^{2}} + C

Solution steps

\int (a^{2} + b^{2} x^{2})^{\frac{1}{2}} x d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{1}{2 b ^{2}} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = a^{2} + b^{2} x^{2}

= \frac{1}{2 b ^{2}} \cdot \frac{2}{3} (a^{2} + b^{2} x^{2})^{\frac{3}{2}}

Simplify

\frac{1}{2 b ^{2}} \cdot \frac{2}{3} (a^{2} + b^{2} x^{2})^{\frac{3}{2}} : \frac{( a ^{2} + b ^{2} x ^{2} ) ^{\frac{3}{2}}}{3 b ^{2}}

= \frac{( a ^{2} + b ^{2} x ^{2} ) ^{\frac{3}{2}}}{3 b ^{2}}

Add a constant to the solution

= \frac{( a ^{2} + b ^{2} x ^{2} ) ^{\frac{3}{2}}}{3 b ^{2}} + C

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What is the integral of (a^2+b^2x^2)^{1/2}x ?
The integral of (a^2+b^2x^2)^{1/2}x is ((a^2+b^2x^2)^{3/2})/(3b^2)+C