What is the integral of 3x^2\sqrt[3]{(5x^3+4)^5} ?

The integral of 3x^2\sqrt[3]{(5x^3+4)^5} is 3/40 (5x^3+4)^{8/3}+C

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integral of 3x^2\sqrt[3]{(5x^3+4)^5}

\frac{3}{40} (5 x^{3} + 4)^{\frac{8}{3}} + C

Solution steps

Take the constant out:

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

Apply u-substitution

= 3 \cdot \int \frac{u ^{\frac{5}{3}}}{15} d u

Take the constant out:

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

= 3 \cdot \frac{1}{15} \cdot \int u^{\frac{5}{3}} d u

Apply the Power Rule

= 3 \cdot \frac{1}{15} \cdot \frac{3}{8} u^{\frac{8}{3}}

Substitute back

u = 5 x^{3} + 4

= 3 \cdot \frac{1}{15} \cdot \frac{3}{8} (5 x^{3} + 4)^{\frac{8}{3}}

Simplify

3 \cdot \frac{1}{15} \cdot \frac{3}{8} (5 x^{3} + 4)^{\frac{8}{3}} : \frac{3}{40} (5 x^{3} + 4)^{\frac{8}{3}}

= \frac{3}{40} (5 x^{3} + 4)^{\frac{8}{3}}

Add a constant to the solution

= \frac{3}{40} (5 x^{3} + 4)^{\frac{8}{3}} + C

What is the integral of 3x^2\sqrt[3]{(5x^3+4)^5} ?
The integral of 3x^2\sqrt[3]{(5x^3+4)^5} is 3/40 (5x^3+4)^{8/3}+C