Question

Find the value of \(\int_4^5 \,log⁡x \,dx\).

(a) 5 log⁡5-log⁡4+1

(b) 5 log⁡5-4 log⁡4-1

(c) 4 log⁡5-4 log⁡4-1

(d) 5-4 log⁡4-log⁡5

I got this question by my college director while I was bunking the class.

This question is from Fundamental Theorem of Calculus-1 topic in division Integrals of Mathematics – Class 12

Answer 1

The correct answer is (b) 5 log⁡5-4 log⁡4-1

For explanation I would say: Let I=\(\int_4^5 \,log⁡x \,dx\).

F(x)=∫ log⁡x dx

By using the formula \(\int \,u.v dx=u \int v \,dx-\int u'(\int \,v \,dx)\), we get

\(\int log ⁡x \,dx=log⁡x \int \,dx-\int(log⁡x)’\int \,dx\)

F(x)=x log⁡x-∫ dx=x(log⁡x-1).

Applying the limits using the fundamental theorem of calculus, we get

I=F(5)-F(4)=(5 log⁡5-5)-(4 log⁡4-4)

=5 log⁡5-4 log⁡4-1.