What is the integral of 5xsqrt(x^2+3) ?

The integral of 5xsqrt(x^2+3) is 5/3 (x^2+3)^{3/2}+C

integral of 5xsqrt(x^2+3)

\int 5 x x^{2} + 3 d x

\frac{5}{3} (x^{2} + 3)^{\frac{3}{2}} + C

Solution steps

\int 5 x x^{2} + 3 d x

Take the constant out:

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

= 5 \cdot \int x x^{2} + 3 d x

Apply u-substitution

= 5 \cdot \int u^{2} d u

Apply the Power Rule

= 5 \cdot \frac{u ^{3}}{3}

Substitute back

u = x^{2} + 3

= 5 \cdot \frac{( x ^{2} + 3 ) ^{3}}{3}

Simplify

5 \cdot \frac{( x ^{2} + 3 ) ^{3}}{3} : \frac{5}{3} (x^{2} + 3)^{\frac{3}{2}}

= \frac{5}{3} (x^{2} + 3)^{\frac{3}{2}}

Add a constant to the solution

= \frac{5}{3} (x^{2} + 3)^{\frac{3}{2}} + C

\int \frac{cos ( e ^{- x} )}{e ^{x}} d x

\int e^{e^{x}} d e^{e^{x}}

\int \frac{1}{4 + 5 cos ^{2} ( x )} d x

\int \frac{5}{6 + 8 x - 4 x ^{2}} d x

\int 9 + 7 x^{2} d x

What is the integral of 5xsqrt(x^2+3) ?
The integral of 5xsqrt(x^2+3) is 5/3 (x^2+3)^{3/2}+C