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.