Solve ∫ (from 0 to pi) of (6sin(θ)-5cos(θ))dθ

$$\int _ { 0 } ^ { \pi } ( 6 \sin ( \theta ) - 5 \cos ( \theta ) ) d \theta$$

Evaluate

$$12$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 6\sin(\theta )-5\cos(\theta )\mathrm{d}\theta$$

Integrate the sum term by term.

$$\int 6\sin(\theta )\mathrm{d}\theta +\int -5\cos(\theta )\mathrm{d}\theta$$

Factor out the constant in each of the terms.

$$6\int \sin(\theta )\mathrm{d}\theta -5\int \cos(\theta )\mathrm{d}\theta$$

Use \int \sin(\theta )\mathrm{d}\theta =-\cos(\theta ) from the table of common integrals to obtain the result. Multiply 6 times -\cos(\theta ).

$$-6\cos(\theta )-5\int \cos(\theta )\mathrm{d}\theta$$

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

$$-6\cos(\theta )-5\sin(\theta )$$

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.

$$-6\cos(\pi )-5\sin(\pi )-\left(-6\cos(0)-5\sin(0)\right)$$

Simplify.

$$12$$