What is the integral of 4x^3(4x^4+5) ?

The integral of 4x^3(4x^4+5) is 2x^8+5x^4+C

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integral of 4x^3(4x^4+5)

\int 4 x^{3} (4 x^{4} + 5) d x

2 x^{8} + 5 x^{4} + C

Solution steps

\int 4 x^{3} (4 x^{4} + 5) d x

Take the constant out:

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

= 4 \cdot \int x^{3} (4 x^{4} + 5) d x

Apply Integration By Parts

: \frac{1}{4} x^{4} (4 x^{4} + 5) - \int 4 x^{7} d x

= 4 (\frac{1}{4} x^{4} (4 x^{4} + 5) - \int 4 x^{7} d x)

\int 4 x^{7} d x = \frac{x ^{8}}{2}

= 4 (\frac{1}{4} x^{4} (4 x^{4} + 5) - \frac{x ^{8}}{2})

Simplify

4 (\frac{1}{4} x^{4} (4 x^{4} + 5) - \frac{x ^{8}}{2}) : 2 x^{8} + 5 x^{4}

= 2 x^{8} + 5 x^{4}

Add a constant to the solution

= 2 x^{8} + 5 x^{4} + C

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

\int 5 x^{4} - 3 x^{2} + 4 d x

\int \frac{1}{( x + 1 ) x} d x

\int \frac{x}{2 + x - x ^{2}} d x

\int \frac{t ^{2}}{( 1 + t ^{2} ) ^{2}} d t

What is the integral of 4x^3(4x^4+5) ?
The integral of 4x^3(4x^4+5) is 2x^8+5x^4+C