I can't find the integral for

(tanx)^(6)*(secx)^(2)

I tried splitting up tanx into (tanx)^2*(tanx)^4 and let the latter equal (secx)^2 - 1.

Please help, thanks!

To solve the integral of (tanx)^(6)*(secx)^(2), we can use the technique of u-substitution.

Let's start by letting u = tanx, and therefore, du = sec^2(x) dx.

Now, we can rewrite the integral in terms of u as follows:

∫ (tanx)^6 * (secx)^2 dx = ∫ u^6 * du

This simplifies the integral significantly.

Now, we can integrate the new expression:

∫ u^6 * du

To find the antiderivative, we add 1 to the exponent and divide by the new exponent:

= (u^7) / 7 + C

Finally, substitute u back in terms of x:

= (tanx)^7 / 7 + C

So, the integral of (tanx)^(6)*(secx)^(2) is (tanx)^7 / 7 + C, where C is the constant of integration.

To find the integral of (tanx)^6 * (secx)^2, let's use u-substitution.

1. Let u = tan(x).
This implies that du/dx = sec^2(x).
So, du = sec^2(x) dx.

Now, let's rewrite the integral in terms of u:
∫ (tan(x))^6 * (sec(x))^2 dx = ∫ (u^6) * du.

2. Integrate the new expression.
∫ (u^6) * du = (u^7)/7 + C, where C is the constant of integration.

3. Substitute u back in terms of x.
(u^7)/7 + C = (tan(x)^7)/7 + C.

Therefore, the integral of (tan(x))^6 * (sec(x))^2 is (tan(x)^7)/7 + C, where C is the constant of integration.