Solve ∫ ((1+sqrt{x})^2-(1+x)^2) wrt x

$$\int ( ( 1 + \sqrt { x } ) ^ { 2 } - ( 1 + x ) ^ { 2 } ) d x$$

Evaluate

$$-\frac{x^{3}}{3}-\frac{x^{2}}{2}+\frac{4x^{\frac{3}{2}}}{3}+С$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\sqrt{x}\right)^{2}.

$$\int 1+2\sqrt{x}+\left(\sqrt{x}\right)^{2}-\left(1+x\right)^{2}\mathrm{d}x$$

Calculate \sqrt{x} to the power of 2 and get x.

$$\int 1+2\sqrt{x}+x-\left(1+x\right)^{2}\mathrm{d}x$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+x\right)^{2}.

$$\int 1+2\sqrt{x}+x-\left(1+2x+x^{2}\right)\mathrm{d}x$$

To find the opposite of 1+2x+x^{2}, find the opposite of each term.

$$\int 1+2\sqrt{x}+x-1-2x-x^{2}\mathrm{d}x$$

Subtract 1 from 1 to get 0.

$$\int 2\sqrt{x}+x-2x-x^{2}\mathrm{d}x$$

Combine x and -2x to get -x.

$$\int 2\sqrt{x}-x-x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2\sqrt{x}\mathrm{d}x+\int -x\mathrm{d}x+\int -x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int \sqrt{x}\mathrm{d}x-\int x\mathrm{d}x-\int x^{2}\mathrm{d}x$$

Rewrite \sqrt{x} as x^{\frac{1}{2}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{2}}\mathrm{d}x with \frac{x^{\frac{3}{2}}}{\frac{3}{2}}. Simplify. Multiply 2 times \frac{2x^{\frac{3}{2}}}{3}.

$$\frac{4x^{\frac{3}{2}}}{3}-\int x\mathrm{d}x-\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply -1 times \frac{x^{2}}{2}.

$$\frac{4x^{\frac{3}{2}}}{3}-\frac{x^{2}}{2}-\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply -1 times \frac{x^{3}}{3}.

$$\frac{4x^{\frac{3}{2}}}{3}-\frac{x^{2}}{2}-\frac{x^{3}}{3}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{4x^{\frac{3}{2}}}{3}-\frac{x^{2}}{2}-\frac{x^{3}}{3}+С$$

Differentiate w.r.t. x

$$-x^{2}-x+2\sqrt{x}$$