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Trigonometric integrals: The integrals that follow can be solved by using algebra to write the integrands in the form f'(u(x))u'(x)so thatu-substitution will apply.

Solve sec4xtan3xdxby using the Pythagorean identity role="math" localid="1649174356968" tan2x+1=sec2xto rewrite the integrand as (tan2x+1)tan3xsec2x and then applying substitution with u=tanx.

Short Answer

Expert verified

The value of integral is sec4xtan3xdx=sec6x6+sec4x4+C.

Step by step solution

01

Step 1. Given Information 

Solve sec4xtan3xdxby using the Pythagorean identity tan2x+1=sec2xto rewrite the integrand as (tan2x+1)tan3xsec2x and then applying substitution with u=tanx.

02

Step 2. The given integral is ∫sec4xtan3xdx

We can write as

sec4xtan3xdx=(tan2x+1)tan3xsec2xdx

Let

u=tanxdudx=sec2xdu=sec2xdx

03

Step 3. Now the integral is 

sec4xtan3xdx=(u2+1)u3dusec4xtan3xdx=(u2·u3+1·u3)dusec4xtan3xdx=(u5+u3)dusec4xtan3xdx=u5du+u3dusec4xtan3xdx=u5+15+1+u3+13+1+Csec4xtan3xdx=u66+u44+Csec4xtan3xdx=sec6x6+sec4x4+C

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