What is the integral of x/((7x^2+3)^5) ?

The integral of x/((7x^2+3)^5) is -1/(56(7x^2+3)^4)+C

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integral of x/((7x^2+3)^5)

\int \frac{x}{( 7 x ^{2} + 3 ) ^{5}} d x

- \frac{1}{56 ( 7 x ^{2} + 3 ) ^{4}} + C

Solution steps

\int \frac{x}{( 7 x ^{2} + 3 ) ^{5}} d x

Apply u-substitution

= \int \frac{1}{14 u ^{5}} d u

Take the constant out:

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

= \frac{1}{14} \cdot \int \frac{1}{u ^{5}} d u

Apply the Power Rule

= \frac{1}{14} (- \frac{1}{4 u ^{4}})

Substitute back

u = 7 x^{2} + 3

= \frac{1}{14} (- \frac{1}{4 ( 7 x ^{2} + 3 ) ^{4}})

Simplify

\frac{1}{14} (- \frac{1}{4 ( 7 x ^{2} + 3 ) ^{4}}) : - \frac{1}{56 ( 7 x ^{2} + 3 ) ^{4}}

= - \frac{1}{56 ( 7 x ^{2} + 3 ) ^{4}}

Add a constant to the solution

= - \frac{1}{56 ( 7 x ^{2} + 3 ) ^{4}} + C

y^{^{'''}} + 49 y^{^{'}} = 0

t an g e n t e^{x^{5}}

\frac{d}{d x} (8 x^{2} + 7)

\int_{0}^{2} 3^{- x} d x

y^{^{''}} - 2 y^{^{'}} + 10 y = e^{x} cos (3 x)

What is the integral of x/((7x^2+3)^5) ?
The integral of x/((7x^2+3)^5) is -1/(56(7x^2+3)^4)+C