i have to integrate using u substitution, but i am not sure if i did it correct.
�çtan^3(5x)sec^2(5x)dx
u=5x
du=5dx
1/5du=dx
1/5�çtan^3(u)sec^2(u)du
1/5tanusec^2u+c
1/5tan(5x)sec^2(5x)+c
To check if you performed the u-substitution correctly, you can differentiate your answer and see if it matches the original integrand.
In this case, you started with the integrand:
∫ tan^3(5x) sec^2(5x) dx
Then, you made the substitution:
u = 5x
Taking the derivative of both sides, you get:
du/dx = 5
Solving for dx, you get:
dx = du/5
Now, substitute the expression for dx into the original integral:
∫ tan^3(5x) sec^2(5x) dx = ∫ tan^3(u) sec^2(u) (du/5)
Now, simplify the expression:
1/5 ∫ tan^3(u) sec^2(u) du
This is correct so far. Next, you need to integrate the expression with respect to u. The integral of tangent cubed can be solved using a trigonometric identity. According to the identity:
∫ tan^3(u) sec^2(u) du = (1/2) tan^2(u) + C
Where C is the constant of integration.
Now, substitute back the original variable:
1/5 ∫ tan^3(u) sec^2(u) du = 1/5 [ (1/2) tan^2(u) ] + C
Finally, replace u with the original expression:
(1/10) tan^2(5x) + C
This is the final result of the integral. To check if it's correct, you can differentiate it with respect to x and see if it matches the original integrand: tan^3(5x) sec^2(5x).