Solve ∫ (from pi to 1.3 pi) of (sinx+cosx) wrt x

$$\int _ { \pi } ^ { 1.3 \pi } ( \sin x + \cos x ) d x$$

Evaluate

$$\frac{\sqrt{10-2\sqrt{5}}-\sqrt{5}-5}{4}\approx -1.221231742$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

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

$$-\cos(x)+\int \cos(x)\mathrm{d}x$$

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

$$-\cos(x)+\sin(x)$$

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.

$$-\cos(1.3\pi )+\sin(1.3\pi )-\left(-\cos(\pi )+\sin(\pi )\right)$$

Simplify.

$$\frac{\sqrt{2\sqrt{5}-2}\sqrt[4]{5}-5-\sqrt{5}}{4}$$