"Theoretical and conceptual tool" is probably closer to what happened than "used directly", but how "integral" it was is probably in the eye of the beholder. Joukowsky's 1906 model was an influential early part of gradually developing and intertwined theoretical and experimental work with feedback going in both directions. Joukowsky's airfoil itself was too idealized for direct use, but, together with abundant experimental data, it became the basis of more refined models that were used in design more directly, such as Munk's thin airfoil model of 1921. Munk's model itself uses Fourier series ansatz for the rather than complex analysis for the vortex distribution, but...

"The vortex distribution function... is Glauert's approximation and is based on Joukowski transformation results ($A_0$ term) which mainly covers the effect of angle of attack, plus a Fourier series variation ($A_n$ terms) to account for camber. It automatically obeys the Kutta condition with zero vorticity at the trailing edge. It is based on a mapped angular position $θ$ rather than an exact surface location $s$ to allow for ease of integration." [Auld-Srinivas, "2-D Thin Aerofoil Theory. ]

Here is from Highlights From The History of Airfoil Development by Jones:

"In 1902, W. M. Kutta calculated the lift of a thin, cambered plate at zero angle of attack and obtained a substantial lift force without drag in a frictionless fluid. In 1906, a theory of airfoils having rounded leading edges and varying angles of attack was developed by N. E. Joukowski... Beginning in 1921, NACA began collecting airfoil data from laboratories around the world and presenting them in a uniform notation... During this period, the design of airfoils was largely intuitive and based on the experience gained in testing numerous variations. While the experimenters were thus occupied, attempts were being made to extend the theory of Kutta and Joukowski to cover a greater variety of shapes. Thus, R. von Mises, a well-known aerodynamicist, was able to extend Joukowski's theory to cover an infinite series of profiles, all obtained by transformation of a circle. Also, H. Glauert, and T. von Karman obtained generalizations of the Joukowski airfoils.

A new direction in experimental technique appeared when Max Munk of NACA Langley proposed the variable density wind tunnel (see NACA TN 60, 1921)... Soon after the variable density tunnel, Munk developed his "thin airfoil theory." Although less accurate, this theory was a great simplification of earlier theories and permitted the relations between shape, pressure, and lift to be seen more clearly than before. With thin airfoil theory it was relatively easy to design airfoils that had a fixed or even a stable center of pressure. A systematic series of these, known as the M sections, were tested in the variable density tunnel at Langley in the 1920s. Of these, the M-6 and the M-12 found application; for example, on the Waco Taperwing. I used the M-12 on a small racing airplane (Pobjoy Phantom, Fig. 7) which flew in the 1930 National Air Races. Another important result of the thin airfoil theory, brought out by Glauert, is the magnifying effect of a plain trailing-edge flap when used as a control surface."

"Theoretical and conceptual tool" is probably closer to what happened than "used directly", but how "integral" it was is probably in the eye of the beholder. Joukowsky's 1906 model was an influential early part of gradually developing and intertwined theoretical and experimental work with feedback going in both directions, see Highlights From The History of Airfoil Development by Jones. 

Joukowsky's airfoil itself was too idealized for direct use, but, together with abundant experimental data, it became the basis of more refined models that were used in design more directly, such as Munk's thin airfoil model of 1921 elaborated by Glauert. The thin airfoil model manipulates a Fourier series ansatz for the vortex distribution rather than uses complex analysis, but...

"The vortex distribution function... is Glauert's approximation and is based on Joukowski transformation results ($A_0$ term) which mainly covers the effect of angle of attack, plus a Fourier series variation ($A_n$ terms) to account for camber. It automatically obeys the Kutta condition with zero vorticity at the trailing edge. It is based on a mapped angular position $θ$ rather than an exact surface location $s$ to allow for ease of integration." [Auld-Srinivas, "2-D Thin Aerofoil Theory. ]

Here is from Jones:

"In 1902, W. M. Kutta calculated the lift of a thin, cambered plate at zero angle of attack and obtained a substantial lift force without drag in a frictionless fluid. In 1906, a theory of airfoils having rounded leading edges and varying angles of attack was developed by N. E. Joukowski... Beginning in 1921, NACA began collecting airfoil data from laboratories around the world and presenting them in a uniform notation... During this period, the design of airfoils was largely intuitive and based on the experience gained in testing numerous variations. While the experimenters were thus occupied, attempts were being made to extend the theory of Kutta and Joukowski to cover a greater variety of shapes. Thus, R. von Mises, a well-known aerodynamicist, was able to extend Joukowski's theory to cover an infinite series of profiles, all obtained by transformation of a circle. Also, H. Glauert, and T. von Karman obtained generalizations of the Joukowski airfoils.

A new direction in experimental technique appeared when Max Munk of NACA Langley proposed the variable density wind tunnel (see NACA TN 60, 1921)... Soon after the variable density tunnel, Munk developed his "thin airfoil theory." Although less accurate, this theory was a great simplification of earlier theories and permitted the relations between shape, pressure, and lift to be seen more clearly than before. With thin airfoil theory it was relatively easy to design airfoils that had a fixed or even a stable center of pressure. A systematic series of these, known as the M sections, were tested in the variable density tunnel at Langley in the 1920s. Of these, the M-6 and the M-12 found application; for example, on the Waco Taperwing. I used the M-12 on a small racing airplane (Pobjoy Phantom, Fig. 7) which flew in the 1930 National Air Races. Another important result of the thin airfoil theory, brought out by Glauert, is the magnifying effect of a plain trailing-edge flap when used as a control surface."

"Theoretical and conceptual tool" is probably closer to what happened than "used directly", but how "integral" it was is probably in the eye of the beholder. Joukowsky's 1906 model was an influential early part of gradually developing and intertwined theoretical and experimental work with feedback going in both directions. Joukowsky's airfoil itself was too idealized for direct use, but, together with abundant experimental data, it became the basis of more refined models that were used in design more directly, such as Munk's thin airfoil model of 1921.

Here is from Highlights From The History of Airfoil Development by Jones:

"In 1902, W. M. Kutta calculated the lift of a thin, cambered plate at zero angle of attack and obtained a substantial lift force without drag in a frictionless fluid. In 1906, a theory of airfoils having rounded leading edges and varying angles of attack was developed by N. E. Joukowski... Beginning in 1921, NACA began collecting airfoil data from laboratories around the world and presenting them in a uniform notation... During this period, the design of airfoils was largely intuitive and based on the experience gained in testing numerous variations. While the experimenters were thus occupied, attempts were being made to extend the theory of Kutta and Joukowski to cover a greater variety of shapes. Thus, R. von Mises, a well-known aerodynamicist, was able to extend Joukowski's theory to cover an infinite series of profiles, all obtained by transformation of a circle. Also, H. Glauert, and T. von Karman obtained generalizations of the Joukowski airfoils.

A new direction in experimental technique appeared when Max Munk of NACA Langley proposed the variable density wind tunnel (see NACA TN 60, 1921)... Soon after the variable density tunnel, Munk developed his "thin airfoil theory." Although less accurate, this theory was a great simplification of earlier theories and permitted the relations between shape, pressure, and lift to be seen more clearly than before. With thin airfoil theory it was relatively easy to design airfoils that had a fixed or even a stable center of pressure. A systematic series of these, known as the M sections, were tested in the variable density tunnel at Langley in the 1920s. Of these, the M-6 and the M-12 found application; for example, on the Waco Taperwing. I used the M-12 on a small racing airplane (Pobjoy Phantom, Fig. 7) which flew in the 1930 National Air Races. Another important result of the thin airfoil theory, brought out by Glauert, is the magnifying effect of a plain trailing-edge flap when used as a control surface."

"Theoretical and conceptual tool" is probably closer to what happened than "used directly", but how "integral" it was is probably in the eye of the beholder. Joukowsky's 1906 model was an influential early part of gradually developing and intertwined theoretical and experimental work with feedback going in both directions. Joukowsky's airfoil itself was too idealized for direct use, but, together with abundant experimental data, it became the basis of more refined models that were used in design more directly, such as Munk's thin airfoil model of 1921. Munk's model itself uses Fourier series ansatz for the rather than complex analysis for the vortex distribution, but...

"The vortex distribution function... is Glauert's approximation and is based on Joukowski transformation results ($A_0$ term) which mainly covers the effect of angle of attack, plus a Fourier series variation ($A_n$ terms) to account for camber. It automatically obeys the Kutta condition with zero vorticity at the trailing edge. It is based on a mapped angular position $θ$ rather than an exact surface location $s$ to allow for ease of integration." [Auld-Srinivas, "2-D Thin Aerofoil Theory. ]

Here is from Highlights From The History of Airfoil Development by Jones:

"In 1902, W. M. Kutta calculated the lift of a thin, cambered plate at zero angle of attack and obtained a substantial lift force without drag in a frictionless fluid. In 1906, a theory of airfoils having rounded leading edges and varying angles of attack was developed by N. E. Joukowski... Beginning in 1921, NACA began collecting airfoil data from laboratories around the world and presenting them in a uniform notation... During this period, the design of airfoils was largely intuitive and based on the experience gained in testing numerous variations. While the experimenters were thus occupied, attempts were being made to extend the theory of Kutta and Joukowski to cover a greater variety of shapes. Thus, R. von Mises, a well-known aerodynamicist, was able to extend Joukowski's theory to cover an infinite series of profiles, all obtained by transformation of a circle. Also, H. Glauert, and T. von Karman obtained generalizations of the Joukowski airfoils.

A new direction in experimental technique appeared when Max Munk of NACA Langley proposed the variable density wind tunnel (see NACA TN 60, 1921)... Soon after the variable density tunnel, Munk developed his "thin airfoil theory." Although less accurate, this theory was a great simplification of earlier theories and permitted the relations between shape, pressure, and lift to be seen more clearly than before. With thin airfoil theory it was relatively easy to design airfoils that had a fixed or even a stable center of pressure. A systematic series of these, known as the M sections, were tested in the variable density tunnel at Langley in the 1920s. Of these, the M-6 and the M-12 found application; for example, on the Waco Taperwing. I used the M-12 on a small racing airplane (Pobjoy Phantom, Fig. 7) which flew in the 1930 National Air Races. Another important result of the thin airfoil theory, brought out by Glauert, is the magnifying effect of a plain trailing-edge flap when used as a control surface."