Solve phi ( x ) = ∫ (from 5 pi / 4 to x) of (3sint+4cost)dt

$$\phi ( x ) = \int _ { 5 \pi / 4 } ^ { x } ( 3 \sin t + 4 \cos t ) d t$$

Evaluate

$$\frac{8\sin(x)-6\cos(x)+\sqrt{2}}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3\sin(t)+4\cos(t)\mathrm{d}t$$

Integrate the sum term by term.

$$\int 3\sin(t)\mathrm{d}t+\int 4\cos(t)\mathrm{d}t$$

Factor out the constant in each of the terms.

$$3\int \sin(t)\mathrm{d}t+4\int \cos(t)\mathrm{d}t$$

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

$$-3\cos(t)+4\int \cos(t)\mathrm{d}t$$

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

$$-3\cos(t)+4\sin(t)$$

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.

$$-3\cos(x)+4\sin(x)-\left(-3\cos(\frac{1}{4}\times 5\pi )+4\sin(\frac{1}{4}\times 5\pi )\right)$$

Simplify.

$$-3\cos(x)+4\sin(x)+\frac{\sqrt{2}}{2}$$

Differentiate w.r.t. x

$$3\sin(x)+4\cos(x)$$