Question

integrate it plzzzzz

Answer 1

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Answer:

This Integration can be completed by using the indentity that is studied in the chapter definite integrals.

\int\limits^a_0 {f(x)} \, dx =\int\limits^a_0 {f(a-x)} \, dx0∫a​f(x)dx=0∫a​f(a−x)dx

Step-by-step explanation:

I = ∫ log(1 + tanx).dx __________( 1)

I = \int\limits^π/4_0 {log[1+tan(π/4-x)]} \, dx∫π​/40​log[1+tan(π/4−x)]dx

Now let us apply the trigonometric formula for tan(π/4 - x),

We know that,

tan (π/4- x) = [tan45 - tanx]/[1 - tan45 . tanx] = [1 - tanx]/[1 + tanx]

since tan45 = 1

I = \int\limits^π/4_0 {1-[1-tanx]/[1+tanx]} \, dx∫π​/40​1−[1−tanx]/[1+tanx]dx

I = \int\limits^π/4_0 {2/[1+tanx]} \, dx∫π​/40​2/[1+tanx]dx

I = \int\limits^π/4_0 {log2 - log(1+tanx)} \, dx∫π​/40​log2−log(1+tanx)dx

I = \int\limits^π/4_0 {log2} \, dx∫π​/40​log2dx - I

for the value of I from equation ( 1)

2I = log2 . \int\!imits^π/4_0{1}\,dx

2I = log2. π/4

I = π/8 . log2







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