How to Integrate tan^3(x) Using U-Substitution

TL;DR

To integrate tan^3(x), first split it into tan^2(x) and tan(x). Use the identity tan^2(x) = sec^2(x) - 1 to rewrite the integral, then apply u-substitution for the sec^2(x)tan(x) term. The final result is (1/2)tan^2(x) - ln|cos(x)| + C.

Transcript

how can we integrate tangent to the 3rd of xdx what is the indefinite integral of that expression what we're going to do is we're going to split it into two parts tan squared and tangent X now the reason why we want to do that is because tangent squared is equal to an identity 1 plus tangent squared is equal to secant squared so if we subtract both... Read More

Key Insights

• ❎ Tangent squared can be replaced with secant squared minus 1 using the identity 1 plus tangent squared equals secant squared.
• 📏 Integrating secant squared tangent X can be achieved through u-substitution and the power rule.
• 👨‍💼 The integral of tangent X can be computed by converting it into sine over cosine and utilizing u-substitution.
• ➕ The final answer for the integral of tangent cubed X is 1/2 tangent squared X plus ln cosine X plus a constant of integration.

Questions & Answers

Q: How do we integrate secant squared tangent X?

To integrate secant squared tangent X, we can use the power rule and u-substitution. Setting u equal to tangent X, we replace secant squared with du, resulting in the integral of u du. Using the power rule for integration, we add 1 to the exponent and divide by that result, concluding with 1/2 tangent squared X.

Q: What technique can be used to integrate tangent X?

One effective method to integrate tangent X is by converting it into sine over cosine. After using u-substitution and setting u equal to cosine X, we can simplify the expression and arrive at the natural logarithm of cosine X raised to the -1 power, which is equivalent to ln secant X.

Q: Is there an alternative form to the final answer?

Yes, another way to write the final answer for the integral of tangent cubed X is 1/2 tangent squared X + ln cosine X + C. This alternative form removes the negative sign and still represents the same equation.

Summary & Key Takeaways

• In order to integrate tangent to the 3rd of X dx, split the expression into two parts: tangent squared and tangent X.

• Utilize the identity that 1 plus tangent squared is equal to secant squared to replace tangent squared with secant squared minus 1.

• Separate the integral into two parts: the integral of secant squared tangent X and the integral of tangent X.