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Evaluate the integral by making an appropriate substitution. \int \tan x \sec^{4} x dx

Question:

Evaluate the integral by making an appropriate substitution.

{eq}\displaystyle\; \int\tan x \sec^{4} x\,dx {/eq}

Indefinite Integrals :

Solving that indefinite integral where the substitution of the variable is to be made, then the derivative of the variable is applied and then the integral is replaced with the new indefinite integral.

Answer and Explanation:


We have the indefinite integral given as;

{eq}\int\tan x \sec^{4} x\,dx\\ {/eq}

Now, if we take the substitution of:

{eq}u=\sec \left(x\right)\\ {/eq}

we have differentiation as:

{eq}\frac{du}{dx}=\sec \left(x\right)\tan \left(x\right)\\ {/eq}

So the integral will be:

{eq}\int\tan x \sec^{4} x\,dx\\=\int \:u^3du\\ =\frac{u^{3+1}}{3+1}+c\\ =\frac{1}{4}\sec ^4\left(x\right)+C {/eq}


Learn more about this topic:

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Indefinite Integrals as Anti Derivatives

from Math 104: Calculus

Chapter 12 / Lesson 11
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