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Solve 1x2+9dx the following two ways:

(a) with the trigonometric substitution x = 3 tan u;

(b) with algebra and the derivative of the arctangent.

Short Answer

Expert verified

Part (a) The solution of the integral is 13tan-1x3+C.

Part (b) The solution of the integral is 13tan-1x3+C.

Step by step solution

01

Part (a) Step 1. Given Information.

The given integral is1x2+9dx.

02

Part (a) Step 2. Solve. 

We have to solve the integral with the substitution x=3tanu,so the derivative isdx=3sec2udu.

Let's solve the integral by substituting x,

1x2+9dx=13tanu2+93sec2udu=3sec2u9tan2u+9du=3sec2u9tan2u+1du=13sec2utan2u+1duUsetheidentity1+tan2x=sec2x=13sec2usec2udu=131du=13u+CSubstitutebacku,=13tan-1x3+C

03

Part (b) Step 1. Solve. 

We have to solve the integral with algebra and the derivative of the arctangent.

As we know the derivative of the arctangent is 1x2+1.

Now, let's solve the integral,

1x2+9dx=19x29+1dx=191x32+1dxUsethederivativeofarctangent=19×3tan-1x3+C=13tan-1x3+C

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