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Evaluate the indefinite integral:
∫(x²+1)(x³+3x)⁴dx =

User Egze
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Final answer:

To evaluate the indefinite integral ∫(x²+1)(x³+3x)⁴dx, we can expand the expression and simplify the exponents using the power rule for exponents. Then, we can integrate each term separately to find the indefinite integral.

Step-by-step explanation:

To evaluate the given indefinite integral, ∫(x²+1)(x³+3x)⁴dx, we can expand the expression inside the integral using the distributive property:


(x²+1)(x³+3x)⁴ = x²(x³+3x)⁴ + 1(x³+3x)⁴


Next, we can simplify the exponents using the power rule for exponents:


x²(x³+3x)⁴ = x^(2+3(4))(x+x)⁴ = x^14(x⁴+3x)⁴


Now, we can integrate each term separately:


∫x^14(x⁴+3x)⁴ dx = ∫x^18+3x⁵ dx = (1/19)x^19 + (1/4)x^6 + C

To evaluate the indefinite integral ∫(x²+1)(x³+3x)⁴dx, we can expand the expression using the distributive property:

(x²+1)(x³+3x)⁴ = x²(x³+3x)⁴ + 1(x³+3x)⁴

Simplify the exponents:

= x^(2+3(4))(x+x)⁴ = x^14(x⁴+3x)⁴

Now integrate each term separately:

∫x^14(x⁴+3x)⁴ dx = ∫x^18+3x⁵ dx = (1/19)x^19 + (1/4)x^6 + C

Therefore, the result is (1/19)x^19 + (1/4)x^6 + C, where C is the constant of integration.

User Tijmen
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