Definite Integration
General
Grade 12
Question:
The value of $\int_{0}^{1} \frac{x \tan^{-1} x}{(1+x^{2})^{3/2}} dx$ is
\frac{4+\pi}{4\sqrt{2}}
\frac{4-\pi}{4\sqrt{2}}
\frac{\pi}{2}
-\frac{\pi}{2}
Step-by-Step Solution
Key Concept: General
Let $\tan^{-1} x = t \implies \frac{dx}{1+x^2} = dt$<br>$\therefore I = \int_0^{\pi/4} \frac{t \tan t \cdot dt}{\sqrt{1+\tan^2 t}} = \int_0^{\pi/4} t \cdot \sin t dt = -\frac{\pi}{4\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{4-\pi}{4\sqrt{2}}$
Correct Answer: B