Question

How would you evaluate $\int \tan ^{10} x \sec ^{2} x d x ?$

   How would you evaluate $\int \tan ^{10} x \sec ^{2} x d x ?$

Calculus: Early Transcendentals

Calculus: Early Transcendentals

William Briggs, Lyle… 3rd Edition

Chapter 8, Problem 7

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Solved by Expert Amit Srivastava

05/19/2020

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Then, the derivative of $u$ with respect to $x$ is $\frac{du}{dx}=\sec^2 x$. Show more…

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How would you evaluate $\int \tan ^{10} x \sec ^{2} x d x ?$

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Key Concepts

Trigonometric Integration

Trigonometric integration focuses on evaluating integrals that involve trigonometric functions. It often requires using trigonometric identities and considering the derivatives of trigonometric functions to rewrite the integral into a more manageable form.

Integration of Power Functions

Once an integral has been transformed into a simpler form via substitution, it often reduces to the integration of power functions. This involves applying basic power rules for integration, which state that the integral of u raised to a power is u raised to that power plus one, divided by that exponent plus one, provided the exponent is not equal to -1.

u-Substitution

This is a method used to simplify integrals by substituting a part of the integrand with a new variable so that the integral becomes easier to evaluate. In problems where the derivative of a function inside the integrand is also present, u-substitution streamlines the process by turning a complicated expression into one that involves a straightforward power of u.

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