What is the antiderivative of the following expression?
3x(x^2 + 7)^3
hard way: expand the product:
3x^7 + 63x^5 + 441x^3 + 1029x
integral is
3/8 x^8 + 21/2 x^6 + 441/4 x^4 + 1029/2 x^2
easy way: let u = x^2+7
du = 2x dx
Integral of 3/2 u^3 du
is
3/8 u^4 = 3/8 (x^2+7)^4
To find the antiderivative of the expression 3x(x^2 + 7)^3, we can use the power rule of integration and the chain rule.
Step 1: Rewrite the expression using the power rule.
- The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration.
Applying the power rule, we have the expression:
∫3x(x^2 + 7)^3 dx
Step 2: Apply the chain rule.
- To deal with the nested expression (x^2 + 7)^3, we apply the chain rule of differentiation in reverse.
- Recall that the chain rule states d(u^n)/dx = n(u^(n-1)) * du/dx.
To apply the chain rule, we consider u = x^2 + 7.
- Differentiating u with respect to x, we get du/dx = 2x.
Now, we substitute u and du/dx back into the expression:
∫3x(x^2 + 7)^3 dx
= ∫3x(u)^3 dx
= ∫3u^3 * (du/dx) dx
= ∫3u^3 * (2x) dx
Step 3: Simplify the expression and integrate.
We can simplify the expression ∫3u^3 * (2x) dx as follows:
- The constant factors 3 and 2 can be combined to give 6.
- The antiderivative of u^3 is (u^4)/4.
Putting it all together, we have:
∫3u^3 * (2x) dx = 6 ∫(u^4)/4 dx
= 6 * [(u^4)/4] + C
= 3/2 * (x^2 + 7)^4 + C
Therefore, the antiderivative of the expression 3x(x^2 + 7)^3 is 3/2 * (x^2 + 7)^4 + C, where C is the constant of integration.