Solve (2^xe^x+x)dx

$$( 2 ^ { x } e ^ { x } + x ) d x$$

Differentiate w.r.t. x

$$d\left(x\left(\left(\ln(2)+1\right)\times \left(2e\right)^{x}+2\right)+\left(2e\right)^{x}\right)$$

Steps Using Derivative Rule for Sum

Use the distributive property to multiply 2^{x}e^{x}+x by d.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(2^{x}e^{x}d+xd\right)x)$$

Use the distributive property to multiply 2^{x}e^{x}d+xd by x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2^{x}e^{x}dx+dx^{2})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$2dx^{2-1}+d\times \left(2e\right)^{x}x^{1-1}$$

Subtract 1 from 2.

$$2dx^{1}+d\times \left(2e\right)^{x}x^{1-1}$$

Subtract 1 from 1.

$$2dx^{1}+d\times \left(2e\right)^{x}x^{0}$$

For any term t, t^{1}=t.

$$2dx+d\times \left(2e\right)^{x}x^{0}$$

For any term t except 0, t^{0}=1.

$$2dx+d\times \left(2e\right)^{x}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$2dx+d\times \left(2e\right)^{x}$$

Evaluate

$$dx\left(x+\left(2e\right)^{x}\right)$$