Question

let f be the function defined by $f(x) = \int_{2}^{3x^5} \frac{1}{(t^2 + 2t)^6} dt$. find an expression for $f(x)$.

Explanation:

Step1: Recall the Fundamental Theorem of Calculus and Chain Rule

The Fundamental Theorem of Calculus (Part 1) states that if \( F(x)=\int_{a}^{g(x)} f(t)dt \), then \( F'(x) = f(g(x))\cdot g'(x) \), where \( a \) is a constant and \( g(x) \) is a differentiable function. Here, \( f(t)=\frac{1}{(t^{2}+2t)^{6}} \) and \( g(x) = 3x^{5} \).

Step2: Find \( f(g(x)) \)

Substitute \( t = g(x)=3x^{5} \) into \( f(t) \):
\( f(g(x))=\frac{1}{((3x^{5})^{2}+2(3x^{5}))^{6}}=\frac{1}{(9x^{10}+6x^{5})^{6}} \)

Step3: Find \( g'(x) \)

Differentiate \( g(x) = 3x^{5} \) with respect to \( x \) using the power rule \( \frac{d}{dx}(x^{n})=nx^{n - 1} \):
\( g'(x)=3\times5x^{4}=15x^{4} \)

Step4: Apply the Chain Rule

Using the formula \( F'(x)=f(g(x))\cdot g'(x) \), we multiply the results from Step 2 and Step 3:
\( F'(x)=\frac{1}{(9x^{10}+6x^{5})^{6}}\times15x^{4}=\frac{15x^{4}}{(9x^{10}+6x^{5})^{6}} \)
We can factor the denominator: \( 9x^{10}+6x^{5}=3x^{5}(3x^{5} + 2) \), so \( (9x^{10}+6x^{5})^{6}=(3x^{5}(3x^{5}+2))^{6}=3^{6}x^{30}(3x^{5}+2)^{6} \), but the simplified form above is also correct. Alternatively, we can factor out \( 3x^{5} \) from the denominator:
\( F'(x)=\frac{15x^{4}}{(3x^{5}(3x^{5}+2))^{6}}=\frac{15x^{4}}{729x^{30}(3x^{5}+2)^{6}}=\frac{5x^{4}}{243x^{30}(3x^{5}+2)^{6}}=\frac{5}{243x^{26}(3x^{5}+2)^{6}} \) (by subtracting exponents \( x^{4-30}=x^{- 26}=\frac{1}{x^{26}} \)) or we can leave it as \( \frac{15x^{4}}{(9x^{10}+6x^{5})^{6}} \). Another way to factor the denominator: \( 9x^{10}+6x^{5}=3x^{5}(3x^{5}+2) \), so \( (9x^{10}+6x^{5})^{6}=(3x^{5})^{6}(3x^{5}+2)^{6}=729x^{30}(3x^{5}+2)^{6} \), and \( 15x^{4}/729x^{30}=5x^{4}/243x^{30}=5/(243x^{26}) \), so \( F'(x)=\frac{5}{243x^{26}(3x^{5}+2)^{6}} \) or \( \frac{15x^{4}}{(9x^{10}+6x^{5})^{6}} \).

Answer:

\( \boldsymbol{\frac{15x^{4}}{(9x^{10}+6x^{5})^{6}}} \) (or the factored form \( \boldsymbol{\frac{5}{243x^{26}(3x^{5}+2)^{6}}} \))