Integration and trigonometricfunctions

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Homework Statement


Integrate: 3sec3x(3sec3x+2tan3x)dx


Homework Equations





The Attempt at a Solution


Ok I just multiplied out and obviously got the integrals of both to be tan3x +c and 6sec3x+c
My question is does that make my final answer tan3x+6sec3x +c or +2c?

I swear this is the last integration question I will ask. For awhile.
 
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You can lump the constants together and just add a +C.

But, if I were you, I would differentiate my answer to see if is the correct antiderivative.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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