What is the integral of 4x^3(x^4+5)^6 ?

The integral of 4x^3(x^4+5)^6 is 1/7 (x^4+5)^7+C

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integral of 4x^3(x^4+5)^6

\int 4 x^{3} (x^{4} + 5)^{6} d x

\frac{1}{7} (x^{4} + 5)^{7} + C

Solution steps

\int 4 x^{3} (x^{4} + 5)^{6} 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} (x^{4} + 5)^{6} d x

Apply u-substitution

: \int \frac{u ^{6}}{4} d u

= 4 \cdot \int \frac{u ^{6}}{4} d u

Take the constant out:

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

= 4 \cdot \frac{1}{4} \cdot \int u^{6} d u

Apply the Power Rule

: \frac{u ^{7}}{7}

= 4 \cdot \frac{1}{4} \cdot \frac{u ^{7}}{7}

Substitute back

u = x^{4} + 5

= 4 \cdot \frac{1}{4} \cdot \frac{( x ^{4} + 5 ) ^{7}}{7}

Simplify

4 \cdot \frac{1}{4} \cdot \frac{( x ^{4} + 5 ) ^{7}}{7} : \frac{1}{7} (x^{4} + 5)^{7}

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

Add a constant to the solution

= \frac{1}{7} (x^{4} + 5)^{7} + C

x \to + 1 + lim (1 - x^{2})

\int \frac{arctan ( 7 x )}{1 + 49 x ^{2}} d x

x \to \frac{π}{3} lim (1 - 2 cos (x))

\frac{5}{x}, at x = 4

\frac{d y}{d x} = \frac{3 x + 2 y + 1}{3 x + 2 y - 1}

What is the integral of 4x^3(x^4+5)^6 ?
The integral of 4x^3(x^4+5)^6 is 1/7 (x^4+5)^7+C