Solve ∫ (from 0 to pi) of (7sin(t)+8cos(t))dt

$$\int _ { 0 } ^ { \pi } ( 7 \sin ( t ) + 8 \cos ( t ) ) d t$$

Evaluate

$$14$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 7\sin(t)+8\cos(t)\mathrm{d}t$$

Integrate the sum term by term.

$$\int 7\sin(t)\mathrm{d}t+\int 8\cos(t)\mathrm{d}t$$

Factor out the constant in each of the terms.

$$7\int \sin(t)\mathrm{d}t+8\int \cos(t)\mathrm{d}t$$

Use \int \sin(t)\mathrm{d}t=-\cos(t) from the table of common integrals to obtain the result. Multiply 7 times -\cos(t).

$$-7\cos(t)+8\int \cos(t)\mathrm{d}t$$

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

$$-7\cos(t)+8\sin(t)$$

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.

$$-7\cos(\pi )+8\sin(\pi )-\left(-7\cos(0)+8\sin(0)\right)$$

Simplify.

$$14$$