What is the integral of-4x(-6x^2-1)^6 ?

The integral of-4x(-6x^2-1)^6 is -1/21 (6x^2+1)^7+C

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integral of-4x(-6x^2-1)^6

\int - 4 x (- 6 x^{2} - 1)^{6} d x

- \frac{1}{21} (6 x^{2} + 1)^{7} + C

Solution steps

\int - 4 x (- 6 x^{2} - 1)^{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 (- 6 x^{2} - 1)^{6} d x

Factor

- 6 x^{2} - 1 : - (6 x^{2} + 1)

= - 4 \cdot \int x (- (6 x^{2} + 1))^{6} d x

(- (6 x^{2} + 1))^{6} = (6 x^{2} + 1)^{6}

= - 4 \cdot \int x (6 x^{2} + 1)^{6} d x

Apply u-substitution

: \int \frac{u ^{6}}{12} d u

= - 4 \cdot \int \frac{u ^{6}}{12} d u

Take the constant out:

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

= - 4 \cdot \frac{1}{12} \cdot \int u^{6} d u

Apply the Power Rule

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

= - 4 \cdot \frac{1}{12} \cdot \frac{u ^{7}}{7}

Substitute back

u = 6 x^{2} + 1

= - 4 \cdot \frac{1}{12} \cdot \frac{( 6 x ^{2} + 1 ) ^{7}}{7}

Simplify

- 4 \cdot \frac{1}{12} \cdot \frac{( 6 x ^{2} + 1 ) ^{7}}{7} : - \frac{1}{21} (6 x^{2} + 1)^{7}

= - \frac{1}{21} (6 x^{2} + 1)^{7}

Add a constant to the solution

= - \frac{1}{21} (6 x^{2} + 1)^{7} + C

\frac{d}{d x} (x^{2} - \frac{2}{3} x^{3})

\frac{\partial}{\partial x} (x e^{x})

\frac{d}{d x} (2 x^{2} e^{\frac{x}{2}})

\frac{( s - 2 )}{s ^{2} - 2 s + 10}

x \to - 5 lim (\frac{1}{1 + 5})

What is the integral of-4x(-6x^2-1)^6 ?
The integral of-4x(-6x^2-1)^6 is -1/21 (6x^2+1)^7+C