To determine

To verify:

∫xa+bx dx=215b23bx-2aa+bx32+C

Answer

∫xa+bx dx=215b23bx-2aa+bx32+C is correct

Explanation

1)  Concept:i.

ddx∫f(x)dx=fx

ii. By applying the product, the power and the sum or the difference rules of differentiation, simplify.

2) Calculations:

Consider,

215b23bx-2aa+bx32+C

Taking derivative with respect to x,

ddx215b23bx-2aa+bx32+C

Applying the product, the power, the chain and the sum rule of differentiation,

=215b2a+bx32·ddx3bx-2a+3bx-2addxa+bx32+0

=215b2a+bx32·3b+3bx-2a·32a+bx12·ddxa+bx

=215b23ba+bx32+3bx-2a·32a+bx12·b

=a+bx2·3b15b2a+bx+215b23bx-2a·32b

After simplification,

=a+bx2a5b+25x+3x5-2a5b

=a+bx2x5+3x5=xa+bx

Therefore,

∫xa+bx dx=215b23bx-2aa+bx32+C

Hence, the formula has been verified.

Conclusion:We have verified that

∫xa+bx dx=215b23bx-2aa+bx32+C