The value of \(\int^\pi_0 x^3 \sin xdx\) is

Concept:

Integration by parts: Integration by parts is a method to find integrals of products

• The formula for integrating by parts is given by,
• ∫u v dx = u∫v dx −∫u' (∫v dx) dx

Where u is the function u(x) and v is the function v(x)

ILATE Rule: Usually, the preference order of this rule is based on some functions such as Inverse, Logarithm, Algebraic, Trigonometric and Exponent.

Calculation:

Let I = \(\int^\pi_0 x^3 \sin xdx\)

Apply by parts rule, we get

\(\rm =x^3 \int^\pi_0sinxdx- \int^\pi_03x^2(-cosx)dx\)

\(\rm =[x^3(-cosx)]_0^\pi+3[x^2\int^\pi_0cosxdx- \int^\pi_02x(sinx)dx]_0^\pi\)

\(\rm =\pi^3+0-6\int^\pi_0x(sinx)dx\)

\(\rm=\pi^3-6[x\int^\pi_0sinxdx- \int^\pi_0(-cosx)dx]\)

\(\rm =\pi^3-6[\pi- 0]\)

= π3 - 6π