If f0=0, f3=3 and f^(′) 3 =4, then the value of ∫ 0 1 xf^(″) 3x dx is equal to

Correct answer: 1. Let, Ι= ∫ 0 1 xf^(″) 3x dx Put 3x=t⇒dx=dt/3 Ι=1/9 ∫ 0 3 tf^(″) t dt Ι=1/9 tf^(′) t 0 3 − ∫ 0 3 f^(′) t dt =1/9 3f^(′) 3 − f t 0 3 =1/93×4-3=1

Q20 / 116 · NTA Abhyas (JEE Main) 2020 · Mathematics · Definite Integration

If $f (0) = 0, f (3) = 3$ and $f^{'} (3) = 4,$ then the value of $\int_{0}^{1} x f^{″} (3 x) d x$ is equal to

Correct answer: 1

Solution

Let,

Ι = \int_{0}^{1} x f^{″} (3 x) d x

Put

3 x = t \Rightarrow d x = \frac{d t}{3}

Ι = \frac{1}{9} \int_{0}^{3} t f^{″} (t) d t

Ι = \frac{1}{9} [{[t f^{'} (t)]}_{0}^{3} - \int_{0}^{3} f^{'} (t) d t]

= \frac{1}{9} [3 f^{'} (3) - {(f (t))}_{0}^{3}]

= \frac{1}{9} (3 \times 4 - 3) = 1

Hence, the correct answer is 1.

If f0=0, f3=3 and f ′ 3 =4, then the value of ∫ 0 1 x f ″ 3x dx is equal to

Solution

More Definite Integration Questions

If $f (0) = 0, f (3) = 3$ and $f^{'} (3) = 4,$ then the value of $\int_{0}^{1} x f^{″} (3 x) d x$ is equal to