Definite Integration
Definite Integration
nta_pyq_2025_apr
Grade 12
Question:
Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x) = \displaystyle\int_0^x tf(t)\,dt$. If $g(x^3) = x^6+x^7$, then value of $\displaystyle\sum_{r=1}^{15}f(r^3)$ is:
Step-by-Step Solution
Key Concept: Differentiate $g(x^3) = x^6+x^7$ with respect to $x$ using the chain rule to get $g'(x^3)\cdot 3x^2 = 6x^5+7x^6$, then use $g'(u) = uf(u)$ to extract $f(u)$.
Since $g(x) = \int_0^x tf(t)dt$, by FTC $g'(x) = xf(x)$.
Differentiating $g(x^3) = x^6+x^7$:
$$g'(x^3)\cdot 3x^2 = 6x^5+7x^6 \Rightarrow x^3 f(x^3)\cdot 3x^2 = 6x^5+7x^6.$$
$$3x^5 f(x^3) = 6x^5+7x^6 \Rightarrow f(x^3) = 2+\frac{7x}{3}.$$
So $f(r^3) = 2+\dfrac{7r}{3}$.
$$\sum_{r=1}^{15}f(r^3) = 2(15)+\frac{7}{3}\cdot\frac{15\cdot 16}{2} = 30+280 = 310.$$
Correct Answer: 4