Question

Find the integration of sinx+ x2​

Answer 1

Answer:

Answer:

( {x}^{3} \div 3 )+ - \cos(x) + (5 {x}^{4} \div 4) + c \: (x

3

÷3)+−cos(x)+(5x

4

÷4)+c

where c is a constant

Answer 2

Answer:

x² cos x + 2x sin x + 2cosx + C where C = constant of integration

Proof:

Let I = ∫x². sinx dx

We will evaluate the integral using Method of Integration by Parts the formula for which is

∫u d(v)=u.v-∫v d(u) where u and v are functions of x. To do this , we write

I = ∫x². d(-cos x) = - ∫x². d(cos x)

Take u = x² and v = cos x. Then

I = - [x². cos x - ∫cos x . d(x²)] + C1= - x². cos x + ∫cos x . 2x dx + C1

= - x². cos x + 2∫x cos x dx + C1 = - x² cos x + 2∫x. d(sin x)

We again integrate by parts to get

I = - x². cos x + 2 x sinx - 2∫sin x.dx + C1 + C2 ( C1, C2 are constants of integration)

= - x². cos x + 2x.sin x + 2cosx + C (Proved)

where C = C1 + C2 is the final constant of integration.

Verification of result:

I = - x². cos x + 2x.sin x + 2cosx + C

Differentiating

dI/dx = d/dx(- x². cos x + 2x.sin x + 2cosx + C ) = - d/dx (x². cos x) + 2.d/dx (x.sin x) + 2d/dx (cos x) + 0

= -2x cos x + x² sin x + 2x cos x + 2sin x -2sin x

Cancellation of equal terms gives

dI/dx = x² sin x, the integrand or the given function.

Hence the result is verified to be correct.

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