Solve ∫ (from 5 to 10) of x+sin(x)+1.2 wrt x

$$\int_{ 5 }^{ 10 } x+ \sin ( x ) +1.2 d x$$

Evaluate

$$\frac{2\cos(5)+87-2\cos(10)}{2}\approx 44.622733715$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x+\sin(x)+1.2\mathrm{d}x$$

Integrate the sum term by term.

$$\int x\mathrm{d}x+\int \sin(x)\mathrm{d}x+\int 1.2\mathrm{d}x$$

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}.

$$\frac{x^{2}}{2}+\int \sin(x)\mathrm{d}x+\int 1.2\mathrm{d}x$$

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

$$\frac{x^{2}}{2}-\cos(x)+\int 1.2\mathrm{d}x$$

Find the integral of 1.2 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{2}}{2}-\cos(x)+\frac{6x}{5}$$

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.

$$\frac{10^{2}}{2}-\cos(10)+1.2\times 10-\left(\frac{5^{2}}{2}-\cos(5)+1.2\times 5\right)$$

Simplify.

$$\frac{1}{2}\left(-2\cos(10)+87+2\cos(5)\right)$$