What is the integral of 2x^{2x}(ln(x)+1) ?

The integral of 2x^{2x}(ln(x)+1) is x^{2x}+C

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integral of 2x^{2x}(ln(x)+1)

\int 2 x^{2 x} (ln (x) + 1) d x

x^{2 x} + C

Solution steps

\int 2 x^{2 x} (ln (x) + 1) d x

Take the constant out:

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

= 2 \cdot \int x^{2 x} (ln (x) + 1) d x

Apply u-substitution

: \int \frac{1}{2} d u

= 2 \cdot \int \frac{1}{2} d u

Integral of a constant:

\int a d x = a x

= 2 \cdot \frac{1}{2} u

Substitute back

u = x^{2 x}

= 2 \cdot \frac{1}{2} x^{2 x}

Simplify

2 \cdot \frac{1}{2} x^{2 x} : x^{2 x}

= x^{2 x}

Add a constant to the solution

= x^{2 x} + C

\frac{d}{d x} (e^{\frac{- x}{y}})

\int_{2}^{5} \frac{1}{x + 1} d x

\frac{\partial}{\partial x} (\frac{x}{x ^{2} + 2 y ^{2}})

\int \frac{81 - x ^{2}}{x ^{2}} d x

y = 3 x^{4} - 12 x^{3}

What is the integral of 2x^{2x}(ln(x)+1) ?
The integral of 2x^{2x}(ln(x)+1) is x^{2x}+C