Solve ∫ (from 0 to 1) of (1/2x^2+5sqrt{x}+cosx) wrt x

$$\int _ { 0 } ^ { 1 } ( \frac { 1 } { 2 } x ^ { 2 } + 5 \sqrt { x } + \cos x ) d x$$

Evaluate

$$\frac{2\sin(1)+7}{2}\approx 4.341470985$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{x^{2}}{2}+5\sqrt{x}+\cos(x)\mathrm{d}x$$

Integrate the sum term by term.

$$\int \frac{x^{2}}{2}\mathrm{d}x+\int 5\sqrt{x}\mathrm{d}x+\int \cos(x)\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\frac{\int x^{2}\mathrm{d}x}{2}+5\int \sqrt{x}\mathrm{d}x+\int \cos(x)\mathrm{d}x$$

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 \frac{1}{2} times \frac{x^{3}}{3}.

$$\frac{x^{3}}{6}+5\int \sqrt{x}\mathrm{d}x+\int \cos(x)\mathrm{d}x$$

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 5 times \frac{2x^{\frac{3}{2}}}{3}.

$$\frac{x^{3}}{6}+\frac{10x^{\frac{3}{2}}}{3}+\int \cos(x)\mathrm{d}x$$

Use \int \cos(x)\mathrm{d}x=\sin(x) from the table of common integrals to obtain the result.

$$\frac{x^{3}}{6}+\frac{10x^{\frac{3}{2}}}{3}+\sin(x)$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{1^{3}}{6}+\frac{10}{3}\times 1^{\frac{3}{2}}+\sin(1)-\left(\frac{0^{3}}{6}+\frac{10}{3}\times 0^{\frac{3}{2}}+\sin(0)\right)$$

Simplify.

$$\frac{1}{2}\left(7+2\sin(1)\right)$$