Find the indefinite integral and check the result by differentiation.?

Question

int x^3(x^4+3)^2dx

Answer 1

int x^3(x^4+3)^2dx = int x^3(x^8+6x^4+9)dx
= int (x^11+6x^7+9x^3) dx
= int x^11dx + 6int x^7dx + 9int x^3dx
= (x^12)/12 + 6/8 x^8 + 9/4 x^4

Answer 2

First simplify:

int x^11+6x^7+9x^3 dx

Then int:

1/12x^12+3/4x^8+9/4x^4+C

Check the answer by differentiating it.

Answer 3

??(x) + (a million/(6?x)) dx initiate up by utilising technique of writing all radicals as exponents. undergo in techniques ?x = x^(a million/2) additionally a million/x = x?¹ so (a million/6?x) = a million/ 6x^(a million/2) = (a million/6)x^?½ Now the crucial is ?x^½ + (a million/6)x^?½ dx Integrals concerning addition would perhaps be broken into separate integrals: ?x^½ + (a million/6)x^?½ dx = ?x^½dx + ?(a million/6)x^?½ dx = (2/3)x^(3/2) + (a million/3)x^½ + C Now attempt by utilising technique of taking the by utilising-product: f(x) = (2/3)x^(3/2) + (a million/3)x^½ + C f '(x) = (3/2)*(2/3)x^(3/2 - a million) + (a million/2)*(a million/3)x^(a million/2 - a million) + 0 f '(x) = a million*x^½ + (a million/6)x^?½ = ?x + (a million/6?x) < -the unique integrand.

Answer 4

∫x^3(x^4+3)^2dx
= ∫(1/4)(x^4+3)^2d (x^4+3)
= (1/12)(x^4+3)^3 + C

Check
[(1/12)(x^4+3)^3 + C]' = (3/12)*(x^4+3)^2 * 4x^3 = x^3(x^4+3)^2
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Ideas: Use reversed chain rule instead of expanding it.

Answer 5

The easiest way is to expand the terms so that you have polynomial. Integration will be a snap then.