integral of 5cos(x)-4sin(x) — step-by-step solution

1) Let's compute the integral:

\int (5 \cdot \cos (x) - 4 \cdot \sin (x)) d x

For the math expression:

5 \cdot \cos (x)

, since (sin(a))' = a'*cos(a), we try differentiating

\sin (x)

2) Let's differentiate:

\frac{d}{d x} (\sin (x))

ⓘ The derivative of sin(x) is cos(x):

\frac{d}{d x} (\sin (x))

\cos (x)

\cos (x)

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

ⅰ

\int (5 \cdot \cos (x) - 4 \cdot \sin (x)) d x

5 \cdot \sin (x) + \int (- 4 \cdot \sin (x)) d x

5 \cdot \sin (x) + \int (- 4 \cdot \sin (x)) d x

For the math expression:

4 \cdot \sin (x)

, since (cos(a))' = -a'*sin(a), we try differentiating

\cos (x)

3) Let's perform the differentiation:

\frac{d}{d x} (\cos (x))

ⓘ The derivative of cos(x) is -sin(x):

\frac{d}{d x} (\cos (x))

- \sin (x)

- \sin (x)

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

ⓘ The derivative of

- (-4 \cdot \cos (x))

- 4 \cdot \sin (x)

, so

- (-4 \cdot \cos (x))

is an antiderivative of

- 4 \cdot \sin (x)

\int (- 4 \cdot \sin (x)) d x

- (-4 \cdot \cos (x))

5 \cdot \sin (x) - (-4 \cdot \cos (x))

ⓘ Extract the minus signs: an even number of minus signs cancels out, an odd number leaves one minus sign:

- (-4 \cdot \cos (x))

4 \cdot \cos (x)

5 \cdot \sin (x) + 4 \cdot \cos (x)

▶Indefinite integrals are defined up to an additive constant C, so this yields:

4 \cdot \cos (x) + 5 \cdot \sin (x) + C