Evaluate the integral given below.
integral(x^2+13)^2dx
lots of ways but brute force:
(x^2 + 13)^2 = x^4 + 26 x^2 + 169
so integrate
x^4 dx + 26 x^2 dx + 169 dx
x^5/5 + 26 (x^3/3) + 169 x + constant
To evaluate the integral ∫(x^2+13)^2 dx, we can use the power rule of integration.
The power rule states that ∫x^n dx = (1/(n+1)) * x^(n+1), where n is any real number except for -1.
In this case, the exponent of the function is 2, so we can rewrite the integral as:
∫(x^2+13)^2 dx = ∫((x^2+13))^2 dx
To simplify, let's expand the square:
= ∫(x^4 + 26x^2 + 169) dx
Now, we can use the power rule to integrate each term separately:
∫x^4 dx = 1/5 * x^5
∫26x^2 dx = 26/3 * x^3
∫169 dx = 169x
Therefore, the integral becomes:
∫(x^2+13)^2 dx = 1/5 * x^5 + 26/3 * x^3 + 169x + C
Where C is the constant of integration.