How to Prepare VITEEE with Google Gemini AI ?

1. Rewrite \tan^3 x:

\tan^3 x = \tan x \cdot \tan^2 x = \tan x (\sec^2 x – 1)

2. Substitute this into the integral:

\int_0^{\pi/4} \tan^3 x, dx = \int_0^{\pi/4} (\tan x \sec^2 x – \tan x), dx

3. Separate the integral:

= \int_0^{\pi/4} \tan x \sec^2 x, dx – \int_0^{\pi/4} \tan x, dx

4. Evaluate the first integral \left(\int_0^{\pi/4} \tan x \sec^2 x, dx\right):

Use the substitution u = \tan x, so du = \sec^2 x, dx. The limits change from x = 0 \implies u = \tan 0 = 0 to x = \pi/4 \implies u = \tan(\pi/4) = 1. \int_0^1 u, du = \left[\frac{u^2}{2}\right]_0^1 = \frac{1^2}{2} – \frac{0^2}{2} = \frac{1}{2}

5. Evaluate the second integral \left(\int_0^{\pi/4} \tan x, dx\right):

\int \tan x, dx = \int \frac{\sin x}{\cos x}, dx. Use the substitution v = \cos x, so dv = -\sin x, dx. The limits change from x = 0 \implies v = \cos 0 = 1 to x = \pi/4 \implies v = \cos(\pi/4) = \frac{1}{\sqrt{2}}. \int_1^{1/\sqrt{2}} -\frac{1}{v}, dv = \left[-\ln |v|\right]_1^{1/\sqrt{2}} = \left(-\ln \left(\frac{1}{\sqrt{2}}\right)\right) – (-\ln 1) = -\ln (2^{-1/2}) – 0 = – \left(-\frac{1}{2} \ln 2\right) = \frac{1}{2} \ln 2

6. Combine the results:

\int_0^{\pi/4} \tan^3 x, dx = \frac{1}{2} – \frac{1}{2} \ln 2 = \frac{1}{2}(1 – \ln 2)

Therefore, the value of the definite integral \int_0^{\pi/4} \tan^3 x, dx is \frac{1}{2}(1 – \ln 2).”