What is the integral of 5xsqrt(x^2+1) ?

The integral of 5xsqrt(x^2+1) is 5/3 (x^2+1)^{3/2}+C

integral of 5xsqrt(x^2+1)

\int 5 x x^{2} + 1 d x

\frac{5}{3} (x^{2} + 1)^{\frac{3}{2}} + C

Solution steps

\int 5 x x^{2} + 1 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} + 1 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} + 1

= 5 \cdot \frac{( x ^{2} + 1 ) ^{3}}{3}

Simplify

5 \cdot \frac{( x ^{2} + 1 ) ^{3}}{3} : \frac{5}{3} (x^{2} + 1)^{\frac{3}{2}}

= \frac{5}{3} (x^{2} + 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{5}{3} (x^{2} + 1)^{\frac{3}{2}} + C

y^{^{''}} - 2 y^{^{'}} - 11 y = 0

d er i v a t i v e s (t) = \frac{5}{t ^{2} + 6 t - 8}

\int \frac{x + 4}{x ^{8}} d x

x \to 0 lim (\frac{1}{e ^{2 x} - 1})

x \to 2 lim (\frac{4 x}{x - 4})

What is the integral of 5xsqrt(x^2+1) ?
The integral of 5xsqrt(x^2+1) is 5/3 (x^2+1)^{3/2}+C