03. Tan^-1 (secx+tan X), Then- If Y Tan (a) 1 (b) 112 Dx S (c) 1 (d) -1​

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Introduction

In this discussion, we will explore the evaluation of the integral tan^-1 (secx+tan x) and then determine the value of y tan (a) 1 (b) 112 dx S (c) 1 (d) -1. The integral in question involves the inverse tangent function and trigonometric functions, which are fundamental concepts in mathematics.

Evaluating the Integral

To evaluate the integral tan^-1 (secx+tan x), we can use the following approach:

  • Let u = secx + tan x. Then, du = (sec x tan x + sec^2 x) dx.
  • We can rewrite the integral as ∫ tan^-1 (u) du.
  • Using the substitution u = secx + tan x, we can rewrite the integral as ∫ tan^-1 (secx + tan x) (sec x tan x + sec^2 x) dx.
  • Expanding the integral, we get ∫ tan^-1 (secx + tan x) sec x tan x dx + ∫ tan^-1 (secx + tan x) sec^2 x dx.
  • Using the substitution u = secx + tan x, we can rewrite the integral as ∫ tan^-1 (u) u du + ∫ tan^-1 (u) du.
  • Evaluating the integrals, we get (1/2) u^2 tan^-1 (u) + (1/2) (u^2 - 1) + C.
  • Substituting back u = secx + tan x, we get (1/2) (secx + tan x)^2 tan^-1 (secx + tan x) + (1/2) ((secx + tan x)^2 - 1) + C.

Determining the Value of y tan (a) 1 (b) 112 dx S (c) 1 (d) -1

To determine the value of y tan (a) 1 (b) 112 dx S (c) 1 (d) -1, we need to evaluate the integral ∫ tan (a) 1 (b) 112 dx.

  • Using the substitution x = a + (b - a) t, we can rewrite the integral as ∫ tan (a + (b - a) t) (b - a) dt.
  • Expanding the integral, we get ∫ tan (a) tan (b - a) t (b - a) dt.
  • Evaluating the integral, we get (1/2) (b - a)^2 log |cos (b - a) t| - (1/2) (b - a)^2 t + C.
  • Substituting back x = a + (b - a) t, we get (1/2) (b - a)^2 log |cos (b - a) (x - a)| - (1/2) (b - a)^2 (x - a) + C.

Conclusion

In conclusion, we have evaluated the integral tan^-1 (secx+tan x) and determined the value of y tan (a) 1 (b) 112 dx S (c) 1 (d) -1. The integral involves the inverse tangent function and trigonometric functions, which are fundamental concepts in mathematics.

Final Answer

The final answer is (d) -1.

Introduction

In our previous discussion, we evaluated the integral tan^-1 (secx+tan x) and determined the value of y tan (a) 1 (b) 112 dx S (c) 1 (d) -1. In this Q&A article, we will address some common questions and provide additional insights into the evaluation of the integral.

Q: What is the main concept behind the evaluation of the integral tan^-1 (secx+tan x)?

A: The main concept behind the evaluation of the integral tan^-1 (secx+tan x) is the use of substitution and trigonometric identities to simplify the integral.

Q: How do you simplify the integral tan^-1 (secx+tan x)?

A: To simplify the integral tan^-1 (secx+tan x), we can use the following steps:

  • Let u = secx + tan x. Then, du = (sec x tan x + sec^2 x) dx.
  • We can rewrite the integral as ∫ tan^-1 (u) du.
  • Using the substitution u = secx + tan x, we can rewrite the integral as ∫ tan^-1 (secx + tan x) (sec x tan x + sec^2 x) dx.
  • Expanding the integral, we get ∫ tan^-1 (secx + tan x) sec x tan x dx + ∫ tan^-1 (secx + tan x) sec^2 x dx.
  • Using the substitution u = secx + tan x, we can rewrite the integral as ∫ tan^-1 (u) u du + ∫ tan^-1 (u) du.
  • Evaluating the integrals, we get (1/2) u^2 tan^-1 (u) + (1/2) (u^2 - 1) + C.
  • Substituting back u = secx + tan x, we get (1/2) (secx + tan x)^2 tan^-1 (secx + tan x) + (1/2) ((secx + tan x)^2 - 1) + C.

Q: What is the final answer to the integral tan^-1 (secx+tan x)?

A: The final answer to the integral tan^-1 (secx+tan x) is (1/2) (secx + tan x)^2 tan^-1 (secx + tan x) + (1/2) ((secx + tan x)^2 - 1) + C.

Q: How do you determine the value of y tan (a) 1 (b) 112 dx S (c) 1 (d) -1?

A: To determine the value of y tan (a) 1 (b) 112 dx S (c) 1 (d) -1, we need to evaluate the integral ∫ tan (a) 1 (b) 112 dx.

  • Using the substitution x = a + (b - a) t, we can rewrite the integral as ∫ tan (a + (b - a) t) (b - a) dt.
  • Expanding the integral, we get ∫ tan (a) tan (b - a) t (b - a) dt.
  • Evaluating the integral, we get (1/2) (b - a)^2 log |cos (b - a) t| - (1/2) (b - a)^2 t + C.
  • Substituting back x = a + (b - a) t, we get (1/2) (b - a)^2 log |cos (b - a) (x - a)| - (1/2) (b - a)^2 (x - a) + C.

Q: What is the final answer to the value of y tan (a) 1 (b) 112 dx S (c) 1 (d) -1?

A: The final answer to the value of y tan (a) 1 (b) 112 dx S (c) 1 (d) -1 is (d) -1.

Conclusion

In conclusion, we have addressed some common questions and provided additional insights into the evaluation of the integral tan^-1 (secx+tan x) and the value of y tan (a) 1 (b) 112 dx S (c) 1 (d) -1. The integral involves the inverse tangent function and trigonometric functions, which are fundamental concepts in mathematics.