What is the integral of x^3sqrt(x^2+43) ?

The integral of x^3sqrt(x^2+43) is 1/5 (x^2+43)^{5/2}-43/3 (x^2+43)^{3/2}+C

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integral of x^3sqrt(x^2+43)

\int x^{3} x^{2} + 43 d x

\frac{1}{5} (x^{2} + 43)^{\frac{5}{2}} - \frac{43}{3} (x^{2} + 43)^{\frac{3}{2}} + C

Solution steps

\int x^{3} x^{2} + 43 d x

Apply u-substitution

: \int (u^{2} - 43) u^{2} d u

= \int (u^{2} - 43) u^{2} d u

Expand

(u^{2} - 43) u^{2} : u^{4} - 43 u^{2}

= \int u^{4} - 43 u^{2} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{4} d u - \int 43 u^{2} d u

\int u^{4} d u = \frac{u ^{5}}{5}

\int 43 u^{2} d u = \frac{43 u ^{3}}{3}

= \frac{u ^{5}}{5} - \frac{43 u ^{3}}{3}

Substitute back

u = x^{2} + 43

= \frac{( x ^{2} + 43 ) ^{5}}{5} - \frac{43 ( x ^{2} + 43 ) ^{3}}{3}

Simplify

\frac{( x ^{2} + 43 ) ^{5}}{5} - \frac{43 ( x ^{2} + 43 ) ^{3}}{3} : \frac{1}{5} (x^{2} + 43)^{\frac{5}{2}} - \frac{43}{3} (x^{2} + 43)^{\frac{3}{2}}

= \frac{1}{5} (x^{2} + 43)^{\frac{5}{2}} - \frac{43}{3} (x^{2} + 43)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{5} (x^{2} + 43)^{\frac{5}{2}} - \frac{43}{3} (x^{2} + 43)^{\frac{3}{2}} + C

ln (x e^{x} + 8)

f (x) = 4 (100)^{7}

\int x^{3} - \frac{1}{2} d x

f (x) = - \frac{1}{( x - 5 ) ^{2}}, at x = - 3

\int x \cdot cos (\frac{π x}{2}) d x

What is the integral of x^3sqrt(x^2+43) ?
The integral of x^3sqrt(x^2+43) is 1/5 (x^2+43)^{5/2}-43/3 (x^2+43)^{3/2}+C