What is the integral of (x^4+3)^{3/2}x^3 ?

The integral of (x^4+3)^{3/2}x^3 is 1/10 (x^4+3)^{5/2}+C

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integral of (x^4+3)^{3/2}x^3

\int (x^{4} + 3)^{\frac{3}{2}} x^{3} d x

\frac{1}{10} (x^{4} + 3)^{\frac{5}{2}} + C

Solution steps

\int (x^{4} + 3)^{\frac{3}{2}} x^{3} d x

Apply u-substitution

: \int \frac{( u + 3 ) ^{\frac{3}{2}}}{4} d u

= \int \frac{( u + 3 ) ^{\frac{3}{2}}}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int (u + 3)^{\frac{3}{2}} d u

Apply u-substitution

: \int v^{\frac{3}{2}} d v

= \frac{1}{4} \cdot \int v^{\frac{3}{2}} d v

Apply the Power Rule

: \frac{2}{5} v^{\frac{5}{2}}

= \frac{1}{4} \cdot \frac{2}{5} v^{\frac{5}{2}}

Substitute back

= \frac{1}{4} \cdot \frac{2}{5} (x^{4} + 3)^{\frac{5}{2}}

Simplify

\frac{1}{4} \cdot \frac{2}{5} (x^{4} + 3)^{\frac{5}{2}} : \frac{1}{10} (x^{4} + 3)^{\frac{5}{2}}

= \frac{1}{10} (x^{4} + 3)^{\frac{5}{2}}

Add a constant to the solution

= \frac{1}{10} (x^{4} + 3)^{\frac{5}{2}} + C

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t \to - \infty lim (e^{t} + 1)

What is the integral of (x^4+3)^{3/2}x^3 ?
The integral of (x^4+3)^{3/2}x^3 is 1/10 (x^4+3)^{5/2}+C