What is the integral of 3xsqrt(1+x^2) ?

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

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integral of 3xsqrt(1+x^2)

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

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

Solution steps

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

Take the constant out:

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

= 3 \cdot \int x 1 + x^{2} d x

Apply u-substitution

: \int u^{2} d u

= 3 \cdot \int u^{2} d u

Apply the Power Rule

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

= 3 \cdot \frac{u ^{3}}{3}

Substitute back

u = 1 + x^{2}

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

Simplify

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

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

Add a constant to the solution

= (1 + x^{2})^{\frac{3}{2}} + C

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