The value of \(\int^\pi_0 x^3 \sin xdx\) is1. π3 - 6π2. -π3 - 6π

1.	The value of \(\int^\pi_0 x^3 \sin xdx\) is1. π3 - 6π2. -π3 - 6π3. -π3 + 6π4. π3 + 6π
Answer» Correct Answer - Option 1 : π³ - 6π 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 Hence, option (1) is correct.

The value of \(\int^\pi_0 x^3 \sin xdx\) is1. π3 - 6π2. -π3 - 6π3. -π3 + 6π4. π3 + 6π

Answer» Correct Answer - Option 1 : π³ - 6π

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

Hence, option (1) is correct.

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