Solve ∫ (from 0 to pi) of (-6sin(t)-6cos(t))dt

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

Evaluate

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Solution Steps

Evaluate the indefinite integral first.

$$\int 6\left(-\sin(t)-\cos(t)\right)\mathrm{d}t$$

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

$$6\left(-\int \sin(t)\mathrm{d}t-\int \cos(t)\mathrm{d}t\right)$$

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

$$6\left(\cos(t)-\int \cos(t)\mathrm{d}t\right)$$

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

$$6\left(\cos(t)-\sin(t)\right)$$

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.

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

Simplify.

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