Solve ∫ (from 0 to pi) of (cosx+3sinx) wrt x

$$\int _ { 0 } ^ { \pi } ( \cos x + 3 \sin x ) d x$$

Evaluate

$$6$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \cos(x)+3\sin(x)\mathrm{d}x$$

Integrate the sum term by term.

$$\int \cos(x)\mathrm{d}x+\int 3\sin(x)\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int \cos(x)\mathrm{d}x+3\int \sin(x)\mathrm{d}x$$

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

$$\sin(x)+3\int \sin(x)\mathrm{d}x$$

Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result. Multiply 3 times -\cos(x).

$$\sin(x)-3\cos(x)$$

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.

$$\sin(\pi )-3\cos(\pi )-\left(\sin(0)-3\cos(0)\right)$$

Simplify.

$$6$$