Solve ∫ (from 0 to frac{ pi) of { 2}}(5cos(x)+3sin(x)-4) wrt x

$$\int_{ 0 }^{ \frac{ \pi }{ 2 } } (5 \cos ( x ) +3 \sin ( x ) -4) d x$$

Evaluate

$$8-2\pi \approx 1.716814693$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 5\cos(x)+3\sin(x)-4\mathrm{d}x$$

Integrate the sum term by term.

$$\int 5\cos(x)\mathrm{d}x+\int 3\sin(x)\mathrm{d}x+\int -4\mathrm{d}x$$

Factor out the constant in each of the terms.

$$5\int \cos(x)\mathrm{d}x+3\int \sin(x)\mathrm{d}x+\int -4\mathrm{d}x$$

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

$$5\sin(x)+3\int \sin(x)\mathrm{d}x+\int -4\mathrm{d}x$$

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

$$5\sin(x)-3\cos(x)+\int -4\mathrm{d}x$$

Find the integral of -4 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$5\sin(x)-3\cos(x)-4x$$

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.

$$5\sin(\frac{1}{2}\pi )-3\cos(\frac{1}{2}\pi )-4\times \frac{1}{2}\pi -\left(5\sin(0)-3\cos(0)-4\times 0\right)$$

Simplify.

$$8-2\pi$$