Hey guys, have another one I hope you can help me with trying to solve this by substitution. If I let u = tan x , than du = sec^2 x dx . Would it be correct that then dx = du / sec^2 x And now at this point I'm lost
There is a little trick to this one:
$\displaystyle \int \sec^3(x)\tan(x)\,dx=\int \sec^2(x)\cdot(\sec(x)\tan(x))\,dx$.
Let $\displaystyle u=\sec(x) \implies \,du=\sec(x)\tan(x)\,dx$
Now the integral becomes: $\displaystyle \int u^2\,du=\frac{1}{3}u^3+C=\color{red}\boxed{\frac{1 }{3}\sec^3(x)+C}$
Hope this makes sense!
Here's something that will help you out in the future:
If we have an integral in the form of $\displaystyle \int \sin^m(x)\cos^n(x)\,dx$, and
n is odd :
m is odd :
- split off a factor of $\displaystyle \cos(x)$
- apply the identity $\displaystyle \cos^2(x)=1-\sin^2(x)$
- apply the substitution $\displaystyle u=\sin(x)$
m is even and n is even :
- split off a factor of $\displaystyle \sin(x)$
- apply the identity $\displaystyle \sin^2(x)=1-\cos^2(x)$
- apply the substitution $\displaystyle u=\cos(x)$
- use $\displaystyle \sin^2(x)=\frac{1}{2}(1-\cos(2x))$ and $\displaystyle \cos^2(x)=\frac{1}{2}(1+\cos(2x))$ to lower the powers of cosine and sine.
If we have an integral in the form of $\displaystyle \int \tan^m(x)\sec^n(x)\,dx$, and
n is even :
m is odd :
- split off a factor of $\displaystyle \sec^2(x)$
- apply the identity $\displaystyle \sec^2(x)=\tan^2(x)+1$
- apply the substitution $\displaystyle u=\tan(x)$
m is even and n is odd :
- split off a factor of $\displaystyle \sec(x)\tan(x)$
- apply the identity $\displaystyle \tan^2(x)=\sec^2(x)-1$
- apply the substitution $\displaystyle u=\sec(x)$
- use $\displaystyle \tan^2(x)=\sec^2(x)-1$ to get the equation ins terms of secant. Then apply reduction formulas.
If we have an integral in the form of $\displaystyle \int \cot^m(x)\csc^n(x)\,dx$, and
n is even :
m is odd :
- split off a factor of $\displaystyle \csc^2(x)$
- apply the identity $\displaystyle \csc^2(x)=\cot^2(x)+1$
- apply the substitution $\displaystyle u=\cot(x)$
m is even and n is odd :
- split off a factor of $\displaystyle \csc(x)\cot(x)$
- apply the identity $\displaystyle \cot^2(x)=\csc^2(x)-1$
- apply the substitution $\displaystyle u=\csc(x)$
- use $\displaystyle \cot^2(x)=\csc^2(x)-1$ to get the equation ins terms of cosecant. Then apply reduction formulas.
Hope this will help you out in future problems!!!