Visual SolutionPYQ 2024 · Jan Shift 2Tricky

Question

Let

f : (0,\infty) \to \mathbb{R}

and

F(x) = \int_0^x t f(t) \, dt

. If

F(x^2) = x^4 + x^5

, then

\sum_{r=1}^{12} f(r^2)

is equal to:

Solution Path

From

F(u) = u^2 + u^{5/2}

, differentiate to get

f(u) = 2 + \frac{5}{2}\sqrt{u}

. Then

f(r^2) = 2 + \frac{5r}{2}

, sum

= 219

01Question Setup

1/4

F(x) = \int_0^x t\,f(t)\,dt

and

F(x^2) = x^4 + x^5

. Find

\sum_{r=1}^{12} f(r^2)

\sum f(r^2) = \;?

02Find f(t)

2/4

Let

u = x^2

F(u) = u^2 + u^{5/2}

. Since

F'(u) = u\,f(u)

, we get

f(u) = 2 + \frac{5}{2}\sqrt{u}

f(u) = 2 + \frac{5}{2}\sqrt{u}

03Compute SumKEY INSIGHT

3/4

f(r^2) = 2 + \frac{5r}{2}

. Sum

= 24 + \frac{5}{2} \cdot 78 = 24 + 195 = 219

\sum = 219

04Final Answer

4/4

\sum_{r=1}^{12} f(r^2) = 219

\boxed{219}

Concepts from this question1 concepts unlocked

★

EASY

Replace a composite expression with a single variable to simplify the integrand

\int f(g(x))\,g'(x)\,dx = \int f(t)\,dt, \quad t = g(x)

Transforms complex integrands into standard forms that can be integrated directly

Trigonometric integralsRational functionsPower rule integrals

Want more practice?