integ(x*cos(x), x, [0,1]) — step-by-step solution

Let's compute the integral:

\int(x \cdot \cos (x))dx

For

x \cdot \cos (x)

, we use integration by parts:

\int(u1 \cdot v2')dx = u1 \cdot v2 + \int(u1' \cdot v2)dx

, with u1 =

x

and v2' =

\cos (x)

Thus,

u1'

1

Let's compute the integral v2':

\int(\cos (x))dx

\sin (x)

Let's compute the integral

u1 \cdot v2 + \int(u1' \cdot v2)dx

x \cdot \sin (x) - \int(\sin (x))dx

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

Let's evaluate over the interval

[0, 1]

(1 \cdot \sin (1) + \cos (1)) - (0 \cdot \sin (0) + \cos (0))

\sin (1)

+ \cos (1)

+ 0 \cdot \sin (0)

- \cos (0)

\sin (1)

+ \cos (1)

+ 0

- 1

\sin (1) + \cos (1) - 1

The approximate value of:

\cos (1) + \sin (1) - 1

is:

\approx 0.5403 + \approx 0.8415 - 1

\approx 1.3818 - 1

\approx 0.3818

▷Result: $\cos (1) + \sin (1) - 1$

▷▷Numeric value:

\approx 0.3818