NDA Maths · Definite Integration

Recovering a Function from Integral Conditions

When an unknown function has parameters and you are given several integral or derivative conditions, each condition becomes one linear equation — solve the system for the parameters.

All Definite Integration notes NDA Maths strategy

Why this matters

A small (3 PYQ) but reliably HARD subtopic. The work is bookkeeping: turn each given condition (a definite integral, a value, a derivative) into an equation in the unknown coefficients, then solve the linear system.

Concept 1 of 1

Solving for coefficients from integral conditions

Intuition

If a function is a combination of known pieces with unknown coefficients (like

Pe^x+Qe^{2x}+Re^{3x}

), then every integral, value, or derivative you are told gives one equation linking P, Q, R. Collect as many equations as unknowns and solve.

Definition

The procedure:

Write the unknown function with its parameters, e.g. $f(x)=Pe^x+Qe^{2x}+Re^{3x}$ .
Convert each condition into an equation: a value $f(0)$ , an integral $\int_0^c f$ , or a derivative $f'(0)=P+2Q+3R$ .
Solve the resulting linear system for the parameters, then answer the specific question asked.

Worked example

Let

f(x)=Ae^x+Be^{2x}

with

f(0)=3

and

\int_0^{\ln 2} f(x)\,dx = \tfrac72

. Find A and B.

$f(0)=A+B=3$ .
$\int_0^{\ln2} f = [Ae^x + \tfrac{B}{2}e^{2x}]_0^{\ln2} = (2A+2B)-(A+\tfrac{B}{2}) = A + \tfrac32 B = \tfrac72$ .
Solve $A+B=3,\ A+\tfrac32 B=\tfrac72$ : subtract to get $\tfrac12 B=\tfrac12\Rightarrow B=1,\ A=2$ .

Answer:

A=2,\ B=1

Practice this conceptself-check · 3 quick reps

Try it yourself

For

f(x)=Pe^x+Qe^{2x}+Re^{3x}

it is found that

P=1,Q=2,R=3

. What is

f'(0)

Practice — Level 1 (3 reps)

Quick reps to lock in the method. Try each, then check.

1.
How many independent conditions to fix 3 unknown coefficients?
2.
For $f=Pe^x+Qe^{2x}+Re^{3x}$ , write $f'(0)$ .
3.
$\int_0^{c} e^{2x}\,dx = ?$

From the bank · past-year question

Example 1Definite IntegrationMODERATE

Consider the following for the items that follow: Let

f(x)=Pe^{x}+Qe^{2x}+Re^{3x}

, where

P

Q

R

are real numbers. Further

f(0)=6

f'(\ln3)=282

and

\int_{0}^{\ln2}f(x)\,dx=11

What is

f'(0)

equal to?

[Q73 · Apr · 2023]

One condition, one equation — match the counts

You need as many independent conditions as unknown parameters. With three unknowns P, Q, R you must extract three equations (e.g. a value, an integral, and a derivative) before the system is solvable.

Drill 2 more on solving for coefficients from integral conditions

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Watch out for (1)

One condition, one equation — match the counts
→ Solving for coefficients from integral conditions

Mastery check — 2 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Definite IntegrationHARD

Consider the following for the items that follow: Let

f(x)=Pe^{x}+Qe^{2x}+Re^{3x}

, where

P

Q

R

are real numbers. Further

f(0)=6

f'(\ln3)=282

and

\int_{0}^{\ln2}f(x)\,dx=11

What is the value of

R

[Q72 · Apr · 2023]

Example 2Definite IntegrationHARD

Consider the following for the items that follow: Let

f(x)=Pe^{x}+Qe^{2x}+Re^{3x}

, where

P

Q

R

are real numbers. Further

f(0)=6

f'(\ln3)=282

and

\int_{0}^{\ln2}f(x)\,dx=11

What is the value of

Q

[Q71 · Apr · 2023]

Drill every past-year question on this subtopic

3 questions from the bank — paginated, with cart and Word-export support.

Drill the 3 questions